MGU Mathematics Series — Algebra I
Part III: Functions, Graphs, and Linear Models
Slope · Intercepts · Forms · Parallel & Perpendicular · Linear Models
Chapter 7 of 14
MGU Mathematics Series  ·  Algebra I  ·  Library Textbook Edition

Chapter 7:
Linear Functions
and Equations

Slope, intercepts, slope-intercept form, point-slope form, standard form, parallel and perpendicular lines, and linear modeling — the complete algebra of straight lines

■ Math Level: Algebra I ■

■ 15 Sections  •  Part III of 5 ■

Follows Chapter 6: The Coordinate Plane and Graphing Relationships  ·  Prepares students for Chapter 8: Systems

Chapter 7 — Front Matter

Purpose and Overview

This chapter is the central chapter on lines in Algebra I. Everything you have studied so far — variables, expressions, equations, inequalities, functions, and graphing — comes together here. Linear functions and equations are among the most important ideas in all of mathematics, not because they are complicated, but because they appear everywhere: in distance and speed, in cost and budgets, in science experiments, in finance, in computing, and in every graph that shows a steady trend.

By the end of this chapter, you will understand what makes a relationship linear, how slope and intercept describe a line completely, and how to move freely between four representations of the same relationship: an equation, a graph, a table of values, and a verbal description. You will learn three forms of the linear equation and understand when each form is useful. You will study special cases — horizontal lines, vertical lines, parallel lines, and perpendicular lines — and you will see how linear models describe real situations.

Opening Question

A rideshare service charges a flat fee of $2.50 when you start a ride, then $1.80 for every mile you travel. If you ride 5 miles, how much will you pay? If you ride 12 miles? If the fare was $12.70, how far did you travel? These questions are not separate problems — they all come from the same linear relationship. By the end of this chapter, you will write the equation for this relationship, build a table of values, draw a graph, and read the slope and intercept directly from the equation.

15Sections
12Objectives
20Key Terms
20+Worked Examples

Learning Objectives

  1. 1Identify whether a relationship is linear from a table, graph, or equation.
  2. 2Recognize and use constant rate of change as the foundation of linearity.
  3. 3Write, interpret, and graph equations in slope-intercept form.
  4. 4Find slope from two points using the slope formula.
  5. 5Find the y-intercept from a graph, table, or equation.
  6. 6Write linear equations from graphs, tables, and two given points.
  7. 7Write and use equations in point-slope form.
  8. 8Recognize and use standard form of a linear equation.
  9. 9Identify and graph horizontal and vertical lines.
  10. 10Determine whether two lines are parallel or perpendicular.
  11. 11Build and interpret linear models for real situations.
  12. 12Recognize and correct the most common linear function mistakes.
Chapter 7 — Contents

Sections at a Glance

Sections 7.1 – 7.2
The Foundation: What Linear Means
What Makes a Relationship Linear · Constant Rate of Change
Section 7.1

What Makes a Relationship Linear

A relationship between two quantities is linear if one quantity changes at a constant rate compared with the other. That is the entire definition. Everything else in this chapter follows from it.

Think about walking at a steady pace of 3 miles per hour. After 1 hour, you have walked 3 miles. After 2 hours, 6 miles. After 3 hours, 9 miles. Each additional hour adds exactly 3 more miles — the rate of change is constant. Now compare that with a car accelerating from a stoplight: each second does not add the same distance, so that relationship is not linear.

Recognizing Linearity in Tables

To check whether a table shows a linear relationship, compute how the output changes for each equal increase in the input. If the change in output is always the same, the relationship is linear.

Example 7.1.1
Checking a table for linearity
  • ProblemDoes the table below show a linear relationship?

    Hours: 1, 2, 3, 4, 5  |  Pay ($): 14, 28, 42, 56, 70
  • ThinkEach time hours increase by 1, compute the change in pay.
  • Solve 28 − 14 = 14 42 − 28 = 14 56 − 42 = 14 70 − 56 = 14 The change in pay is always 14.
  • CheckThe rate of change is constant: $14 per hour for every hour worked.
Answer: Yes. The table shows a linear relationship with a constant rate of change of $14/hr.
Example 7.1.2
A non-linear table — area of a square
  • ProblemSide length (cm): 1, 2, 3, 4, 5  |  Area (cm²): 1, 4, 9, 16, 25. Is this linear?
  • Solve 4 − 1 = 3 9 − 4 = 5 16 − 9 = 7 25 − 16 = 9 Changes are 3, 5, 7, 9 — not constant.
Answer: Not linear. The area of a square is a quadratic relationship.

Recognizing Linearity in Graphs and Equations

A relationship is linear if its graph is a straight line. A curved graph always signals a nonlinear relationship. In equations, a linear equation has variables only to the first power — no x², no √x, no 1/x. It can always be written as y = mx + b.

TypeLinear?Reason
y = 3x + 5✓ YesBoth variables to first power
2x − y = 7✓ YesCan be rewritten as y = 2x − 7
y = x² + 3✗ NoVariable is squared
y = 1/x✗ NoVariable is in the denominator
y = √x✗ NoVariable is under a radical
Student Note

The equation 3x + 2y = 12 looks different from y = −(3/2)x + 6, but they describe the same line. Linearity is defined by structure, not by which form you see. Throughout this chapter you will learn to move freely between forms.

Practice 7.1Concept and Skill Practice

For tables 4–6, show the change in output for each row. State whether the relationship is linear.

  1. In your own words, what does it mean for a relationship to be linear?
  2. How can you test whether a table shows a linear relationship?
  3. What does a linear graph look like on a coordinate plane?
  4. Table: x = 0, 1, 2, 3 and y = 5, 8, 11, 14. Linear or not?
  5. Table: x = 0, 1, 2, 3 and y = 1, 3, 9, 27. Linear or not?
  6. Table: x = 2, 4, 6, 8 and y = 10, 6, 2, −2. Linear or not?
  7. Is y = 4x − 9 linear? Explain.
  8. Is y = x³ + 1 linear? Explain.
  9. Is 5x + 3y = 15 linear? Explain.
  10. Is y = 6/x linear? Explain.
Section 7.2

Constant Rate of Change

The most important property of a linear function is that its rate of change is constant. Rate of change measures how much the output (usually y) changes per unit increase in the input (usually x). For a linear function, this ratio is always the same.

Rate of Change Formula

rate of change = Δy / Δx = (y₂ − y₁) / (x₂ − x₁)

The symbol Δ (delta) means "change in." For a linear function, this ratio is constant no matter which two points you choose.

Example 7.2.1
Rate of change from a table — draining water tank
  • ProblemTime (min): 0, 5, 10, 15, 20  |  Water (gal): 200, 160, 120, 80, 40. Find the rate of change.
  • Solve Rows 1–2: (160 − 200) / (5 − 0) = −40/5 = −8 Rows 2–4: (80 − 160) / (15 − 5) = −80/10 = −8
  • CheckBoth calculations give −8. The relationship is linear.
Answer: Rate of change = −8 gallons per minute. The negative sign means the water level is decreasing.
Example 7.2.2
Writing a rule from a constant rate of change
  • ProblemA printer prints 12 pages per minute. At time 0, it has printed 0 pages. Write a rule for the number of pages after t minutes.
  • ThinkRate of change = 12 pages/min. Starting value = 0.
  • Solvef(t) = 12t
  • CheckAfter 7 minutes: 12 × 7 = 84 pages. ✓
Answer: f(t) = 12t. After 7 minutes, 84 pages have been printed.
Practice 7.2Rate of Change Practice
  1. What is meant by "rate of change"? How is it related to slope?
  2. If a linear function has a negative rate of change, what does the graph look like?
  3. Table: x = 0, 2, 4, 6 and y = 3, 7, 11, 15. Find the rate of change. Is it constant?
  4. Table: x = 1, 3, 5, 7 and y = 10, 4, −2, −8. Find the rate of change.
  5. A hiker climbs from 1,200 ft at 8:00 a.m. to 1,700 ft at 10:00 a.m. to 2,200 ft at noon. Find the rate of change in feet per hour. Is the relationship linear?
  6. A phone plan costs $20/month plus $0.05/text. Write a rule for the monthly cost. What is the rate of change?
Sections 7.3 – 7.11
The Three Forms of a Linear Equation
Slope-Intercept · Graphing · Slope Formula · y-Intercept · Writing Equations · Point-Slope · Standard Form
Section 7.3

Slope-Intercept Form

Slope-Intercept Form

y = mx + b

m = slope (rate of change).   b = y-intercept (the value of y when x = 0).

When an equation is in this form, you can read both key numbers directly — no calculation needed.

Example 7.3.1
Identifying slope and y-intercept
  • ProblemIdentify the slope and y-intercept of each equation.
    (a) y = 3x + 7    (b) y = −½x + 4    (c) y = 5x    (d) y = −6
  • Solve (a) m = 3, b = 7 (b) m = −1/2, b = 4 (c) m = 5, b = 0 (passes through origin) (d) m = 0, b = −6 (horizontal line)
Answer: The coefficient of x is always the slope; the constant is always the y-intercept.
Example 7.3.2
Converting to slope-intercept form
  • ProblemWrite 4x + 2y = 10 in slope-intercept form.
  • ThinkIsolate y by performing inverse operations on both sides.
  • Solve 4x + 2y = 10 2y = −4x + 10 (subtract 4x) y = −2x + 5 (divide every term by 2)
  • CheckSubstitute x = 0: y = 5. Then x = 2: y = 1. Verify in original: 4(2) + 2(1) = 10. ✓
Answer: y = −2x + 5. Slope m = −2, y-intercept b = 5.
Student Warning

When dividing to isolate y, divide every term — including the constant — by the coefficient of y. A common mistake is to divide only the variable terms and leave the constant unchanged.

Practice 7.3Slope-Intercept Form

Identify slope and y-intercept for problems 1–5. Convert to slope-intercept form for problems 6–10.

  1. y = 2x − 5
  2. y = −4x + 1
  3. y = ¾x
  4. y = 7
  5. y = −x + 9
  6. 2x + y = 6
  7. 3x − y = −4
  8. 6x + 3y = 12
  9. −x + 4y = 8
  10. 5x − 2y = 10
Section 7.4

Graphing y = mx + b

Graphing Procedure

Step 1: Identify slope m and y-intercept b.   Step 2: Plot the y-intercept at (0, b).   Step 3: From (0, b), move up or down by the rise, then right by the run. Plot the second point.   Step 4: Connect with a straight line, extend with arrows.   Step 5: Label the line.

Example 7.4.1
Graphing from slope-intercept form — positive slope
  • ProblemGraph y = 2x + 1.
  • Thinkm = 2 = 2/1 (rise 2, run 1). y-intercept b = 1.
  • Solve Plot (0, 1) on the y-axis.
    From (0, 1): move up 2, right 1 → point (1, 3).
    From (1, 3): move up 2, right 1 → point (2, 5).
    Draw the line through all three points.
  • CheckSubstitute x = 2: y = 2(2) + 1 = 5. Point (2, 5) lies on the line. ✓
Answer: A line rising left to right, crossing the y-axis at 1.
Diagram Instruction

Draw a coordinate plane. Mark (0, 1) on the y-axis. Draw a slope triangle from (0, 1) to (1, 3): label the horizontal leg "run = 1" and the vertical leg "rise = 2." Plot (1, 3). Draw a line through (0, 1) and (1, 3) with arrows at both ends. Label the line y = 2x + 1.

Example 7.4.2
Graphing a negative slope
  • ProblemGraph y = −¾x + 5.
  • Thinkm = −3/4: fall 3, run 4. b = 5.
  • SolvePlot (0, 5). From (0, 5): move down 3, right 4 → (4, 2). Draw the line.
  • CheckSubstitute x = 4: y = −¾(4) + 5 = −3 + 5 = 2. ✓
Answer: A line falling from left to right, crossing the y-axis at 5.
Tip — Negative Slopes

When the slope is negative, the rise is negative — you move down. When the slope is a fraction a/b, the numerator is the rise and the denominator is the run. Always move right for the run. You can verify any graph by building a table of three or four points and checking they are collinear.

Practice 7.4Graphing Practice

Graph each equation. Show the y-intercept, the slope triangle, and at least one additional labeled point.

  1. y = x + 3
  2. y = 2x − 4
  3. y = −x + 2
  4. y = ⅓x − 1
  5. y = −⅖x + 4
  6. y = 3
  7. y = −2x
  8. A candle is 10 in. tall and burns 1.5 in./hr. Write an equation for height after x hours, then graph it. What does the y-intercept represent? When does the candle burn out?
Section 7.5

Finding Slope from Two Points

Slope Formula

m = (y₂ − y₁) / (x₂ − x₁)

Subtract in the same order in both numerator and denominator. The labeling of which point is "1" and which is "2" does not matter, as long as you are consistent.

Example 7.5.1
Slope from two points
  • ProblemFind the slope through (2, 5) and (6, 13).
  • Solvem = (13 − 5) / (6 − 2) = 8/4 = 2
  • CheckReverse order: (5 − 13)/(2 − 6) = −8/−4 = 2. ✓
Answer: m = 2
Example 7.5.2
Slope with negative values
  • ProblemFind the slope through (−3, 7) and (1, −1).
  • Solvem = (−1 − 7) / (1 − (−3)) = −8 / (1 + 3) = −8/4 = −2
  • Checkx increases from −3 to 1, y decreases from 7 to −1. Negative slope is correct. ✓
Answer: m = −2
Example 7.5.3 & 7.5.4
Zero slope and undefined slope
  • Zero slopeThrough (−4, 3) and (5, 3): m = (3 − 3)/(5 − (−4)) = 0/9 = 0. Horizontal line.
  • UndefinedThrough (2, −1) and (2, 6): m = (6 − (−1))/(2 − 2) = 7/0 = undefined. Vertical line.
Answer: Horizontal lines have slope 0. Vertical lines have undefined slope.
Student Warning — Zero vs. Undefined

A vertical line has no slope — not a slope of zero, but no slope at all. Zero slope belongs to horizontal lines (flat). Undefined slope belongs to vertical lines (straight up and down). Division by zero is never defined in mathematics.

Practice 7.5Slope Formula Practice

Find the slope through each pair of points.

  1. (0, 0) and (4, 12)
  2. (1, 3) and (5, 11)
  3. (2, 8) and (6, 2)
  4. (−1, −4) and (3, 8)
  5. (−5, 2) and (1, −4)
  6. (0, 7) and (3, 7)
  7. (−2, 5) and (−2, −3)
  8. (−4, −1) and (3, −8)
  9. A pool holds 12,000 gal on Monday and 11,600 gal on Friday (4 days later). Find the rate of change in gal/day.
  10. A car goes from mile marker 40 to mile marker 160 in 2 hours. What does the slope represent?
Section 7.6

Finding the y-Intercept

The y-intercept is the value of y when x = 0. It can be found three ways: (1) read it directly from slope-intercept form, (2) read it from a graph where the line crosses the y-axis, or (3) substitute x = 0 into any linear equation and solve for y.

Example 7.6.1
Finding b from an equation
  • ProblemFind the y-intercept of 3x + 5y = 20.
  • Solve3(0) + 5y = 20 5y = 20 y = 4
Answer: y-intercept is 4. The line crosses the y-axis at (0, 4).
Example 7.6.2
Finding b from slope and one point
  • ProblemA line has slope m = 3 and passes through (2, 11). Find the y-intercept.
  • ThinkUse y = mx + b. Substitute m = 3, x = 2, y = 11. Solve for b.
  • Solve11 = 3(2) + b 11 = 6 + b b = 5
  • Checky = 3x + 5. At x = 2: y = 11. ✓
Answer: y-intercept is 5. The full equation is y = 3x + 5.
Practice 7.6Finding the y-Intercept
  1. y = −5x + 9
  2. 2x + 4y = 16
  3. −3x + y = 7
  4. 6x − 2y = −10
  5. m = 4, passes through (1, 6)
  6. m = −2, passes through (3, 5)
  7. m = ½, passes through (6, 4)
Section 7.7

Writing Linear Equations from Graphs

To write an equation from a graph: (1) identify the y-intercept where the line crosses the y-axis, (2) choose a second clear point and compute the slope using the formula. Then write y = mx + b.

Example 7.7.1
Writing an equation from a graph
  • ProblemA line crosses the y-axis at (0, 3) and passes through (2, 7).
  • Solvem = (7 − 3) / (2 − 0) = 4/2 = 2 y = 2x + 3
  • Checkx = 2: y = 2(2) + 3 = 7. ✓
Answer: y = 2x + 3
Example 7.7.2
Using both intercepts to find slope
  • ProblemA line crosses the x-axis at (4, 0) and the y-axis at (0, −2).
  • Solvem = (−2 − 0) / (0 − 4) = −2/−4 = ½ y = ½x − 2
Answer: y = ½x − 2
Practice 7.7Equations from Graphs

Write the equation of each line. Show the slope calculation.

  1. y-int (0,1), through (3, 7)
  2. y-int (0, −4), through (2, 0)
  3. y-int (0, 5), through (5, 0)
  4. Through (0, 0) and (4, 12)
  5. y-int (0, 8), through (2, 2)
  6. x-int (6, 0), y-int (0, 9)
Section 7.8

Writing Linear Equations from Tables

Strategy: Equation from a Table

Step 1: Verify rate of change is constant.   Step 2: Find slope using any two pairs.   Step 3: Find b using y = mx + b with one point (or read from x = 0 row).   Step 4: Write the equation.   Step 5: Check with a third row.

Example 7.8.1
Equation from a table — x = 0 is included
  • Problemx: 0, 1, 2, 3  |  y: 5, 8, 11, 14
  • Solvem = (8 − 5)/(1 − 0) = 3 b = 5 (from row x = 0) y = 3x + 5
  • Checkx = 3: y = 9 + 5 = 14. ✓
Answer: y = 3x + 5
Example 7.8.2
Equation from a table — x = 0 not included
  • Problemx: 2, 4, 6, 8  |  y: 1, 5, 9, 13
  • Solvem = (5 − 1)/(4 − 2) = 4/2 = 2 Using point (2, 1): 1 = 2(2) + b → 1 = 4 + b → b = −3 y = 2x − 3
  • Checkx = 6: y = 12 − 3 = 9. ✓
Answer: y = 2x − 3
Practice 7.8Equations from Tables

Write the linear equation for each table. Show all work.

  1. x: 0, 1, 2, 3  |  y: −1, 2, 5, 8
  2. x: 1, 3, 5, 7  |  y: 7, 3, −1, −5
  3. x: 3, 5, 7, 9  |  y: 10, 16, 22, 28
Section 7.9

Writing Linear Equations from Two Points

Four-Step Procedure

Step 1: Find slope using the formula.   Step 2: Substitute m and one point into y = mx + b. Solve for b.   Step 3: Write the equation.   Step 4: Verify using the second point.

Example 7.9.1
Equation from two points
  • ProblemWrite the equation through (1, 4) and (3, 10).
  • Solve Step 1: m = (10 − 4)/(3 − 1) = 6/2 = 3 Step 2: 4 = 3(1) + b → b = 1 Step 3: y = 3x + 1
  • Checkx = 3: y = 9 + 1 = 10. ✓
Answer: y = 3x + 1
Example 7.9.2
Fractional slope
  • ProblemWrite the equation through (−2, 7) and (4, 4).
  • Solve m = (4 − 7)/(4 − (−2)) = −3/6 = −½ Using (4, 4): 4 = −½(4) + b → 4 = −2 + b → b = 6 y = −½x + 6
  • Checkx = −2: y = 1 + 6 = 7. ✓
Answer: y = −½x + 6
Practice 7.9Two-Point Equations

Write the equation through each pair. Show all four steps.

  1. (0,3) and (2,9)
  2. (1,5) and (4,14)
  3. (3,7) and (6,1)
  4. (−1,4) and (3,−4)
  5. (−3,−1) and (3,5)
  6. (2,0) and (5,9)
  7. (0,8) and (4,0)
  8. (−4,3) and (2,−3)
Section 7.10

Point-Slope Form

Point-Slope Form

y − y₁ = m(x − x₁)

Use when you know the slope and any one point. Substitute directly — no need to find b first. Convert to slope-intercept form by distributing and isolating y.

Example 7.10.1
Writing in point-slope form
  • ProblemWrite the equation with slope m = 4, passing through (3, 5).
  • Solve y − 5 = 4(x − 3) y − 5 = 4x − 12 y = 4x − 7
  • Checkx = 3: y = 12 − 7 = 5. ✓
Answer: Point-slope: y − 5 = 4(x − 3). Slope-intercept: y = 4x − 7.
Example 7.10.2
Point-slope form with two given points
  • ProblemWrite the equation through (2, −1) and (5, 8).
  • Solve m = (8 − (−1))/(5 − 2) = 9/3 = 3 y − (−1) = 3(x − 2) y + 1 = 3x − 6 y = 3x − 7
  • Checkx = 5: y = 15 − 7 = 8. ✓
Answer: Point-slope: y + 1 = 3(x − 2). Slope-intercept: y = 3x − 7.
Tip — Choosing Which Form to Use

Use slope-intercept form when you know the slope and y-intercept. Use point-slope form when you know the slope and any other point. Use either approach when given two points — both produce the same final equation.

Practice 7.10Point-Slope Form

Write each in point-slope form, then convert to slope-intercept form.

  1. m = 2, through (1, 5)
  2. m = −3, through (4, 7)
  3. m = ½, through (6, 1)
  4. m = −⅔, through (−3, 4)
  5. (1,3) and (4,9)
  6. (−2,5) and (2,−3)
  7. (0,4) and (3,−2)
Section 7.11

Standard Form

Standard Form

Ax + By = C

A, B, and C are integers; A ≥ 0; A and B are not both zero; A, B, C share no common factor. Useful for integer-coefficient work, finding intercepts quickly, and setting up systems of equations.

Example 7.11.1
Converting to standard form
  • ProblemWrite y = 3x − 4 in standard form.
  • Solve y = 3x − 4 −3x + y = −4 (subtract 3x) 3x − y = 4 (multiply by −1)
Answer: 3x − y = 4
Example 7.11.2
Converting a fractional equation to standard form
  • ProblemWrite y = ⅔x − 5 in standard form.
  • Solve Multiply every term by 3: 3y = 2x − 15 −2x + 3y = −15 2x − 3y = 15
Answer: 2x − 3y = 15
Example 7.11.3
Finding intercepts from standard form
  • ProblemFind the x- and y-intercepts of 5x + 2y = 20. Sketch the graph.
  • Solve x-intercept (set y = 0): 5x = 20 → x = 4 → (4, 0) y-intercept (set x = 0): 2y = 20 → y = 10 → (0, 10)
  • SketchPlot both intercepts. Connect with a straight line falling left to right.
Answer: x-intercept (4, 0), y-intercept (0, 10). Line falls from upper left to lower right.
Tip — Intercept Method

Standard form makes finding both intercepts fast: cover up each variable in turn and solve. This intercept method is particularly efficient for graphing — plot two points, draw the line, done.

Practice 7.11Standard Form

Convert to standard form for 1–4. Find x- and y-intercepts for 5–8.

  1. y = 4x + 2
  2. y = −½x + 6
  3. y = ¾x − 3
  4. y = 2x
  5. 3x + y = 9
  6. 4x − 2y = 8
  7. x + 5y = 15
  8. 2x − 6y = −12
Sections 7.12 – 7.13
Special Cases: Horizontal, Vertical, Parallel, and Perpendicular
y = k and x = h · Zero and Undefined Slope · Same Slopes · Negative Reciprocals
Section 7.12

Horizontal and Vertical Lines

Two special cases break the pattern of y = mx + b. A horizontal line has every point at the same y-value. Its equation is y = k for some constant k, and its slope is zero. A vertical line has every point at the same x-value. Its equation is x = h for some constant h, and its slope is undefined.

Line TypeEquation FormSlopeExample
Horizontaly = k0y = 3 (passes through (0,3), (5,3), (−2,3))
Verticalx = hUndefinedx = 4 (passes through (4,0), (4,2), (4,−5))
All othersy = mx + bAny real my = 2x − 1
Example 7.12.1 & 7.12.2
Writing horizontal and vertical line equations
  • HorizontalWrite the equation of the horizontal line through (7, −2). All points on a horizontal line share the same y-value. Answer: y = −2
  • VerticalWrite the equation of the vertical line through (−5, 3). All points on a vertical line share the same x-value. Answer: x = −5
Student Warning — Vertical Lines Are Not Functions

A vertical line fails the definition of a function: it assigns infinitely many y-values to a single x-value. Horizontal lines are functions — each x-value maps to exactly one y-value. This distinction matters whenever you apply the vertical line test to a graph.

Diagram Instructions

Draw two side-by-side coordinate planes. Left: draw a horizontal line at y = 3; label it y = 3; mark the points (−2, 3), (0, 3), (4, 3). Right: draw a vertical line at x = 4; label it x = 4; mark the points (4, −1), (4, 2), (4, 5).

Practice 7.12Horizontal and Vertical Lines
  1. Horizontal through (3, 5)
  2. Vertical through (3, 5)
  3. Horizontal through (−4, 0)
  4. Vertical through (0, −7)
  5. Horizontal with y-intercept −3
  6. Vertical through (2, −8)
  7. y = 10: what type?
  8. x = −2: what type?
  9. 3y = −9: what type?
  10. 2x = 8: what type?
Section 7.13

Parallel and Perpendicular Lines

Parallel Lines

Two lines are parallel if they lie in the same plane and never intersect. Two nonvertical lines are parallel if and only if they have the same slope and different y-intercepts: m₁ = m₂.

Example 7.13.1 & 7.13.2
Identifying and writing parallel lines
  • IdentifyAre y = 3x + 1 and y = 3x − 4 parallel? Both have slope 3; y-intercepts are 1 and −4 (different). Yes — parallel.
  • WriteWrite the line parallel to y = −2x + 5 through (3, 1). Same slope: m = −2. y − 1 = −2(x − 3) y = −2x + 7
Answer: y = −2x + 7. Slopes both −2; y-intercepts 5 and 7 are different. ✓

Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90°). Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals: m₁ · m₂ = −1.

To find the negative reciprocal: flip the fraction and change the sign.

SlopeNegative Reciprocal
2−½
−¾⁴⁄₃
⅕−5
−11
Example 7.13.3 & 7.13.4
Identifying and writing perpendicular lines
  • IdentifyAre y = ⅔x + 1 and y = −³⁄₂x + 4 perpendicular? Check: ⅔ × (−³⁄₂) = −6/6 = −1. ✓ Yes — perpendicular.
  • WriteWrite the line perpendicular to y = 4x − 3 through (8, 2). Perpendicular slope = −¼. y − 2 = −¼(x − 8) y − 2 = −¼x + 2 y = −¼x + 4
  • Check4 × (−¼) = −1. ✓
Answer: y = −¼x + 4
Tip — Horizontal and Vertical Are Always Perpendicular

Horizontal (slope 0) and vertical (undefined slope) lines are always perpendicular to each other. The formula m₁ · m₂ = −1 does not apply here since undefined slope is not a number — simply recognize this case directly.

Practice 7.13Parallel and Perpendicular

Determine parallel, perpendicular, or neither for 1–5. Write the described line for 6–9.

  1. y = 5x + 2 and y = 5x − 7
  2. y = 3x − 1 and y = −⅓x + 4
  3. y = 2x + 1 and y = −2x + 1
  4. y = ⅘x − 3 and y = ⅘x + 1
  5. y = 7x and y = −⅑x + 3
  6. Parallel to y = 3x − 5, through (0, 2)
  7. Perpendicular to y = −2x + 1, through (0, 3)
  8. Parallel to y = −¼x + 6, through (4, 0)
  9. Perpendicular to y = ³⁄₂x, through (3, 5)
Sections 7.14 – 7.15
Real-World Models and Common Mistakes
Building Linear Models · Interpreting Slope and Intercept · Eight Common Errors
Section 7.14

Linear Models in Real Situations

A linear model is a linear equation used to describe a real-world relationship. When building one: (1) identify the two quantities and which is input and which is output; (2) identify the y-intercept (starting value) and slope (rate of change); (3) write the equation; (4) use it to answer questions; (5) interpret the slope and intercept in context with units.

The slope and y-intercept are not just numbers — they carry real meaning. Never report "the slope is 1.80" without also saying what it means: "the fare increases by $1.80 per mile."

Example 7.14.1
Rideshare fare model
  • ProblemA rideshare app charges $2.50 flat plus $1.80/mile. Write a model. Find the fare for 9 miles. Find the distance if the fare is $20.20.
  • DefineLet x = miles traveled. Let y = total fare ($).
  • Solve y = 1.80x + 2.50 9 miles: y = 1.80(9) + 2.50 = 16.20 + 2.50 = $18.70 Fare $20.20: 20.20 = 1.80x + 2.50 17.70 = 1.80x x ≈ 9.8 miles
Answer: 9-mile fare = $18.70. A $20.20 fare ≈ 9.8 miles. Slope: +$1.80/mile. Intercept: $2.50 base fee.
Example 7.14.2
Temperature change model
  • ProblemTemperature is 80°F at noon, falling 3°F/hour. Write a model. When does it reach 50°F?
  • Solve T = −3t + 80 Set T = 50: 50 = −3t + 80 −30 = −3t t = 10 hours
Answer: T = −3t + 80. Temperature reaches 50°F at 10 hours after noon (10:00 p.m.).
Example 7.14.3
Building a model from two data points
  • ProblemA water heater holds 50 gallons when new and 47 gallons after 3 years. Assume linear loss. Write a model. When will it hold less than 40 gallons?
  • Solve Two points: (0, 50) and (3, 47) m = (47 − 50)/(3 − 0) = −3/3 = −1 y = −t + 50 Set y < 40: −t + 50 < 40 −t < −10 t > 10
Answer: y = −t + 50. The heater holds less than 40 gallons after 10 years.
MGU Library Connection
  • MGU Physics — slope as velocity in kinematics; linear equations in motion problems
  • MGU Finance Pages — slope as cost per unit, interest rate, rate of return
  • MGU Statistics — linear regression and lines of best fit in Chapter 14 of this textbook
  • Algebra II Chapter 1 — average rate of change and linear function review
Practice 7.14Linear Modeling Problems

For each problem: define your variable, write the equation, answer all questions, and interpret the slope and y-intercept.

  1. A taxi charges $3.00 base plus $2.50/mile. Write a model. Find the cost for an 8-mile trip. How far for $28?
  2. A student has $240 and withdraws $15/week. Write a model. After how many weeks does the balance fall below $100?
  3. At 6:00 a.m. a car is 30 miles from the city, driving toward it at 60 mph. Write a distance model. When does the car arrive?
  4. Coffee shop customers served: hours 0–3 after 6 a.m. = 5, 22, 39, 56. Write a linear model. Interpret the slope. Predict customers by 10:00 a.m.
  5. At time 0, bacteria = 200; at 2 hours, bacteria = 320. (a) Write a linear model. (b) Is this likely to remain linear long-term? Why or why not?
  6. A plumber charges $75 service fee + $55/hr. Write a model. Cost for 4 hours? How many hours for $350?
  7. At sea level water boils at 100°C. Every 300 meters of elevation lowers boiling point ~1°C. Write a model. At what elevation does water boil at 93°C?
Section 7.15

Common Linear Function Mistakes

Mistake 1 — Confusing slope and y-intercept
In y = 3x + 5, slope = 5 and y-intercept = 3 Correct: coefficient of x is slope (3), constant is y-intercept (5). Always match position, not magnitude.
Mistake 2 — Inconsistent subtraction order in slope formula
m = (y₁ − y₂) / (x₂ − x₁) (mixed order) Correct: subtract in the same order top and bottom. If numerator is y₂ − y₁, denominator must be x₂ − x₁.
Mistake 3 — Forgetting to subtract a negative
x₂ − x₁ where x₁ = −3 written as x₂ − 3 instead of x₂ + 3 Correct: a − (−b) = a + b. Always write the subtraction step explicitly before simplifying.
Mistake 4 — Swapping zero slope and undefined slope
"The vertical line through (4, 2) has slope zero." Correct: horizontal lines have slope 0 (flat). Vertical lines have undefined slope (run = 0, division by zero).
Mistake 5 — Graphing without converting to slope-intercept form
Trying to graph 3x + 2y = 12 directly as slope = 3, intercept = 12 Correct: convert first → y = −(3/2)x + 6, so slope = −3/2 and y-intercept = 6. Or use the intercept method.
Mistake 6 — Sign error when dividing to isolate y
4x − 2y = 10 → y = 4x − 10 (divided only the constant by −2, not the x-term) Correct: −2y = −4x + 10 → y = 2x − 5. Divide every term by the coefficient of y.
Mistake 7 — Perpendicular slope error: opposite instead of negative reciprocal
Perpendicular to slope 3 is slope −3 (just negated, not flipped) Correct: negative reciprocal of 3 is −1/3. Flip the fraction AND change the sign. Check: 3 × (−1/3) = −1. ✓
Mistake 8 — Vertical line equation written as y = ...
Equation of the vertical line through (4, 2): y = 4 Correct: vertical lines have x constant. The equation is x = 4. (y = 4 would be a horizontal line.)
Practice 7.15Error Analysis

Each problem contains an error. Identify it and write the correct solution.

  1. A student finds slope through (1, 5) and (4, 11): m = (4−1)/(11−5) = 3/6 = 1/2. What is wrong?
  2. A student identifies the slope of y = −7x + 4 as 4. What is wrong?
  3. A student writes the vertical line through (−3, 8) as y = −3. What is wrong?
  4. A student says the perpendicular slope to m = 2 is m = −2. What is wrong?
  5. A student converts 4x − 2y = 10 to slope-intercept form and writes y = 4x − 10. What error was made?
  6. A student says a line with slope 0 is vertical. What is wrong?
Chapter 7 — Back Matter

Chapter Summary

Key Terms Review

linear function
A function with a constant rate of change. Its graph is a nonvertical straight line.
linear equation
An equation whose graph is a straight line. All variables appear only to the first power.
slope
The ratio of rise to run: m = (y₂ − y₁)/(x₂ − x₁). Measures steepness and direction of a line.
rate of change
How much the output changes per unit increase in the input. Equal to slope in a linear relationship.
rise
The vertical change between two points on a line (Δy).
run
The horizontal change between two points on a line (Δx).
slope-intercept form
y = mx + b. The slope m and y-intercept b can be read directly from this form.
y-intercept
The point where a graph crosses the y-axis, where x = 0. Written as (0, b).
x-intercept
The point where a graph crosses the x-axis, where y = 0. Written as (a, 0).
point-slope form
y − y₁ = m(x − x₁). Used when slope and any one point are known.
standard form
Ax + By = C, with integer coefficients. Useful for finding intercepts and setting up systems.
constant rate of change
A fixed ratio between changes in output and changes in input. The defining feature of linearity.
horizontal line
A line with equation y = k. Slope is zero. Parallel to the x-axis.
vertical line
A line with equation x = h. Slope is undefined. Not a function.
undefined slope
The slope of a vertical line. The run is zero, making the ratio undefined.
zero slope
The slope of a horizontal line. The rise is zero, making the ratio zero.
parallel lines
Lines in the same plane that never intersect. They have equal slopes and different y-intercepts.
perpendicular lines
Lines that intersect at 90°. Their slopes are negative reciprocals: m₁ · m₂ = −1.
negative reciprocal
Obtained by flipping a fraction and changing its sign. The perpendicular slope of m is −1/m.
linear model
A linear equation used to describe a real-world relationship, with an interpreted slope and y-intercept.

Concept Review Questions

  1. What property must a relationship have to be called linear?
  2. Explain what slope represents in the context of a real linear function. Include units in your answer.
  3. What does the y-intercept tell you about a linear model?
  4. How are slope-intercept form, point-slope form, and standard form related? When is each form most useful?
  5. What is the difference between a line with slope 0 and a line with undefined slope?
  6. How can you determine from two equations whether their graphs are parallel?
  7. How do you find the negative reciprocal of a slope?
  8. Why is a vertical line not a function?
  9. If a line has a negative slope, what does its graph look like?
  10. A model is y = 50 − 6x. What is the starting value? What is the rate of change? Is the output increasing or decreasing?

Skill Practice

Part ASlope Practice
  1. (0,0) and (3,9)
  2. (2,6) and (5,15)
  3. (1,4) and (6,4)
  4. (3,−2) and (3,7)
  5. (−3,5) and (1,−3)
  6. (−6,−4) and (−2,8)
  7. y = 6x − 3
  8. y = −⁵⁄₂x + 4
  9. 4x + y = 9
  10. 3x − 6y = 18
Part BEquation-Writing Practice
  1. Slope 5, y-int −2
  2. Slope −3, y-int 7
  3. m = ⅔, through (3,4)
  4. Through (0,−5) and (2,3)
  5. Through (1,6) and (4,15)
  6. Through (−2,3) and (4,−3)
  7. Through (5,5) and (5,−2)
  8. Parallel to y = 2x + 3, through (0,−4)
  9. Perp. to y = −⅓x + 5, through (0,2)
  10. Horizontal through (−7,4)
Part CGraphing Practice

Label the y-intercept, one additional point, and the line.

  1. y = 4x + 1
  2. y = −3x + 6
  3. y = ½x − 3
  4. y = −⅔x + 4
  5. y = x
  6. y = −5
  7. x = 3
  8. 2x + y = 6
  9. 3x − 2y = 12 (use intercepts)
  10. x − 4y = −8

Word Problems

For each problem: define your variable, write the model, answer all questions, and interpret slope and intercept.

  1. A gym charges $50/month. Write an expression for total cost after any number of months. How much after one full year?
  2. A store sells pens for $1.25 and notebooks for $3.00. Write an expression for total cost of p pens and n notebooks. Find the cost for 4 pens and 3 notebooks.
  3. A car travels 55 mph. Write a distance model for t hours. How far in 3.5 hours?
  4. A factory has startup costs of $4,000 and each unit costs $1.25 to produce. Write a total-cost model. How many units for $8,250?
  5. A field has length 3x meters and width (x + 5) meters. Write a simplified expression for the perimeter.
  6. A business sells candles for $8 each. Materials cost $3/candle; overhead is $200/month. Write a monthly-profit model. Find profit for 60 candles sold.

Error Analysis

Problem 1
Slope through (2, 9) and (5, 3): m = (9−3)/(5−2) = 6/3 = 2

What is the error? What is the correct answer?

Problem 2
Equation with slope 3 through (2,7): y = 3x + 7

What is wrong? Find the correct equation.

Problem 3
6x − 3y = 9 converted to: y = 6x − 9

Identify the error. Write the correct slope-intercept form.

Problem 4
Slope of horizontal line through (4,−5) and (9,−5): "slope is undefined"

What is wrong? What is the correct slope?

Problem 5
Perpendicular to y = 4x − 1 through (0,3): y = 4x + 3

What error was made? Write the correct equation.

Problem 6
A driver is 60 mi from home, driving toward home at 45 mph. Student writes: y = 45x + 60

What is wrong with this model? Write the correct equation and explain.

Chapter 7 Checkpoint

Chapter 7 Checkpoint
Complete without assistance — assesses all major skills from Chapter 7
Part 1 — Concepts
  1. What is the slope of a horizontal line?
  2. What form does the equation of a vertical line take?
  3. In y = −5x + 12, what is the rate of change?
  4. If two lines are perpendicular and one has slope 2/5, what is the slope of the other?
Part 2 — Slope
  1. Find the slope through (−1, −3) and (4, 12).
  2. Find the slope through (3, 7) and (3, −4).
  3. Find the slope through (−5, 2) and (7, 2).
Part 3 — Equations
  1. Write the equation with slope −4 and y-intercept 6.
  2. Write the equation through (1, 3) and (5, 11).
  3. Write the equation through (−3, 8) and (1, 0).
  4. Write in point-slope form: slope 2, through (4, 5). Convert to slope-intercept form.
  5. Write 4x − 2y = 10 in slope-intercept form.
Part 4 — Graphing
  1. Graph y = −³⁄₂x + 4. Label the y-intercept, slope triangle, and one additional point.
  2. Graph 5x + y = 10 using the intercept method.
  3. Graph x = −3 and y = 2 on the same coordinate plane. Label both lines.
Part 5 — Applications
  1. A storage tank holds 500 gallons and loses 40 gallons/day. Write a linear model. After how many days will it hold 180 gallons?
  2. A bike rental costs $8/hr plus a $5 deposit. Write a model. Find the cost for 6 hours. If you paid $33, how many hours did you rent the bike?
  3. Are the lines y = 3x + 4 and 6x − 2y = 8 parallel, perpendicular, or neither? Show your reasoning.

Answers may be placed in the answer key.

Looking Ahead
Chapter 8: Systems of Linear Equations and Inequalities

Chapter 7 worked with one linear equation at a time. A single equation describes a line — all ordered pairs satisfying one condition. Chapter 8 asks: what happens when two conditions must be satisfied at once? That is the situation addressed by a system of linear equations. You will ask where two lines intersect, and find the single ordered pair — or set of values — satisfying both equations simultaneously.

To succeed in Chapter 8, you should be comfortable writing linear equations, graphing lines by slope and y-intercept, recognizing parallel and intersecting lines, and substituting values to check solutions. You will use slope-intercept form, standard form, and all three equation-writing procedures constantly. The three solution methods — graphing, substitution, and elimination — each build directly on skills from this chapter. Systems model situations with two simultaneous constraints: two people traveling toward each other, two quantities that must balance, a break-even point in business. They appear throughout algebra, science, economics, computer science, and everyday decision-making.

MGU Library Connection
  • Linear Equation Forms Reference — Appendix H of this textbook
  • Equation-Solving Strategy Guide — Appendix E of this textbook
  • Function Notation Reference — Appendix G of this textbook
  • Algebra II Chapter 1: Algebra II Readiness and Function Review — average rate of change, transformations
  • Algebra II Chapter 2: Linear Systems, Matrices, and Modeling — extending this chapter's work to three variables and matrices
  • MGU Physics: velocity as slope in kinematics; position-time and velocity-time graphs
  • MGU Finance: cost, revenue, and profit models; break-even analysis
  • MGU Dictionary entries: slope, rate of change, linear function, point-slope form, slope-intercept form, standard form