The Central Idea
Consider four lines:
\[ y=3, \] \[ x=4, \] \[ y=2x+1, \]and
\[ y=-\frac12x+5. \]Each illustrates a different geometric relationship.
- \(y=3\) is horizontal and has slope \(0\).
- \(x=4\) is vertical and has undefined slope.
- \(y=2x+1\) has slope \(2\).
- \(y=-\frac12x+5\) has slope \(-\frac12\).
Because
\[ 2\left(-\frac12\right)=-1, \]the last two lines are perpendicular.
Slope therefore provides a direct algebraic language for describing line direction and geometric relationships.
Horizontal Lines
A horizontal line has the form
\[ \boxed{y=c}, \]where \(c\) is constant.
Every point on the line has the same \(y\)-coordinate.
For example,
\[ y=5 \]contains points such as
\[ (-3,5),\qquad (0,5),\qquad (7,5). \]Slope of a Horizontal Line
Choose any two distinct points on
\[ y=c: \] \[ (x_1,c) \]and
\[ (x_2,c). \]The slope is
\[ m= \frac{c-c}{x_2-x_1} = \frac{0}{x_2-x_1} = \boxed{0}. \]Therefore, every horizontal line has slope \(0\).
Horizontal Lines as Constant Functions
The equation
\[ y=c \]can be written in function notation as
\[ f(x)=c. \]So horizontal lines are exactly the graphs of constant functions over real domains.
They are also affine functions with
\[ m=0. \]Horizontal Lines in Slope-Intercept Form
Write
\[ y=c \]as
\[ y=0x+c. \]Then
\[ \boxed{m=0,\qquad b=c}. \]The slope is zero and the \(y\)-intercept is
\[ \boxed{(0,c)}. \]Vertical Lines
A vertical line has the form
\[ \boxed{x=c}. \]Every point on the line has the same \(x\)-coordinate.
For example,
\[ x=-2 \]contains points such as
\[ (-2,-5),\qquad (-2,0),\qquad (-2,9). \]Slope of a Vertical Line
Choose two distinct points on
\[ x=c: \] \[ (c,y_1) \]and
\[ (c,y_2). \]The slope formula gives
\[ m= \frac{y_2-y_1}{c-c} = \frac{y_2-y_1}{0}. \]Division by zero is undefined.
Therefore, a vertical line has
\[ \boxed{\text{undefined slope}}. \]Undefined Slope Does Not Mean Infinite Slope
It is common to hear a vertical line described informally as having “infinite slope.”
In ordinary algebra, however, its slope is undefined.
The slope formula requires division by zero, and no real number can be assigned as its slope.
Later mathematics can describe vertical directions in other ways, but ordinary slope remains undefined.
Vertical Lines Are Not Functions of \(x\)
A vertical line such as
\[ x=3 \]contains many points with the same \(x\)-coordinate:
\[ (3,-2),\qquad (3,0),\qquad (3,5). \]One input \(x=3\) would correspond to many outputs.
Therefore, a vertical line fails the vertical-line test and cannot be written as a function
\[ y=f(x). \]Vertical Lines in Standard Form
The equation
\[ x=c \]fits standard form
\[ Ax+By=C \]with
\[ A=1,\qquad B=0,\qquad C=c. \]This is one reason standard form is more general than slope-intercept form for representing lines.
Horizontal Versus Vertical
| Feature | Horizontal Line | Vertical Line |
|---|---|---|
| Equation | \(y=c\) | \(x=c\) |
| Slope | \(0\) | Undefined |
| Function of \(x\)? | Yes | No |
| Direction | Left-right | Up-down |
Horizontal and Vertical Lines Are Perpendicular
A horizontal line and a vertical line meet at a right angle.
For example,
\[ y=4 \]and
\[ x=7 \]intersect at
\[ (7,4). \]Their directions are perpendicular.
This is the special zero-slope/undefined-slope case of perpendicular lines.
Parallel Lines
Two distinct lines are parallel if they lie in the same plane and never intersect.
Nonvertical parallel lines have the same slope.
For example,
\[ y=3x+2 \]and
\[ y=3x-5 \]both have slope
\[ \boxed{3}. \]Their graphs have the same direction but different positions.
Why Equal Slopes Give Parallel Lines
Suppose
\[ y=m x+b_1 \]and
\[ y=m x+b_2, \]with
\[ b_1\ne b_2. \]If the lines intersected, there would be some \(x\) for which
\[ mx+b_1=mx+b_2. \]Subtract \(mx\):
\[ b_1=b_2, \]contradicting the assumption that the intercepts differ.
So distinct lines with the same slope do not intersect.
Identical Lines Also Have the Same Slope
Equal slope alone does not prove that two equations represent distinct parallel lines.
Consider
\[ y=2x+5 \]and
\[ 2y=4x+10. \]The second equation simplifies to
\[ y=2x+5. \]These are the same line, not two distinct parallel lines.
Parallel Vertical Lines
Vertical lines do not have finite slopes, so the equal-slope rule cannot be applied directly.
But any two distinct vertical lines are parallel.
For example,
\[ x=2 \]and
\[ x=7 \]never intersect.
They share the same vertical direction.
Parallel Horizontal Lines
Any two distinct horizontal lines are parallel.
For example,
\[ y=-3 \]and
\[ y=8 \]both have slope \(0\).
Because their \(y\)-coordinates differ, they are distinct and never intersect.
Parallel Lines in Slope-Intercept Form
Suppose
\[ L_1:y=m_1x+b_1 \]and
\[ L_2:y=m_2x+b_2. \]For distinct nonvertical lines,
\[ \boxed{ L_1\parallel L_2 \iff m_1=m_2 }. \]If the slopes and intercepts are both equal, the equations represent the same line.
Example: Determine Whether Lines Are Parallel
Compare
\[ y=-\frac34x+2 \]and
\[ y=-\frac34x-9. \]The slopes are
\[ -\frac34 \]and
\[ -\frac34. \]The \(y\)-intercepts differ.
Therefore, the lines are
\[ \boxed{\text{parallel}}. \]Parallel Lines from Standard Form
Consider
\[ A_1x+B_1y=C_1 \]and
\[ A_2x+B_2y=C_2. \]When both lines are nonvertical, their slopes are
\[ -\frac{A_1}{B_1} \]and
\[ -\frac{A_2}{B_2}. \]Parallelism therefore requires
\[ \frac{A_1}{B_1} = \frac{A_2}{B_2}. \]Equivalently, the coefficient pairs are proportional:
\[ \boxed{ (A_1,B_1) \propto (A_2,B_2) }. \]Example from Standard Form
Compare
\[ 2x+3y=12 \]and
\[ 4x+6y=5. \]The coefficient pairs are
\[ (2,3) \]and
\[ (4,6)=2(2,3). \]So the lines have the same direction.
But multiplying the entire first equation by \(2\) would give
\[ 4x+6y=24, \]not \(5\).
Therefore, the lines are distinct and parallel.
Perpendicular Lines
Two lines are perpendicular if they intersect at a right angle:
\[ \boxed{90^\circ}. \]For nonvertical, nonhorizontal lines with slopes \(m_1\) and \(m_2\), perpendicularity is characterized by
\[ \boxed{ m_1m_2=-1 }. \]Negative Reciprocal Slopes
If
\[ m_1\ne0, \]and two nonvertical lines are perpendicular, then
\[ m_2=-\frac{1}{m_1}. \]So the perpendicular slope is the negative reciprocal of the original slope.
For example, if
\[ m_1=3, \]then
\[ \boxed{ m_2=-\frac13 }. \]Why “Negative Reciprocal” Has Two Parts
Suppose
\[ m=\frac25. \]First take the reciprocal:
\[ \frac52. \]Then change the sign:
\[ -\frac52. \]Therefore, the perpendicular slope is
\[ \boxed{-\frac52}. \]Simply changing the sign or simply taking the reciprocal is not enough.
Negative Slope Example
Suppose
\[ m=-\frac47. \]The reciprocal is
\[ -\frac74. \]Changing the sign gives
\[ \boxed{\frac74}. \]Check:
\[ -\frac47\left(\frac74\right) = -1. \]Why Perpendicular Slopes Multiply to \(-1\)
A line with slope
\[ m=\frac{\Delta y}{\Delta x} \]can use the direction vector
\[ (\Delta x,\Delta y). \]A perpendicular direction can be obtained by rotating this vector \(90^\circ\):
\[ (\Delta x,\Delta y) \longmapsto (-\Delta y,\Delta x). \]The new slope is
\[ \frac{\Delta x}{-\Delta y} = -\frac{\Delta x}{\Delta y} = -\frac1m. \]Thus the perpendicular slope is the negative reciprocal.
Dot-Product Derivation
Two nonvertical lines with slopes
\[ m_1 \]and
\[ m_2 \]can use direction vectors
\[ (1,m_1) \]and
\[ (1,m_2). \]Perpendicular vectors have dot product zero:
\[ (1,m_1)\cdot(1,m_2)=0. \]So
\[ 1+m_1m_2=0. \]Therefore,
\[ \boxed{ m_1m_2=-1 }. \]The Horizontal-Vertical Exception
The negative-reciprocal formula assumes finite nonzero slopes.
A horizontal line has slope
\[ 0. \]Its perpendicular line is vertical and has undefined slope.
Because
\[ -\frac10 \]is undefined, this special case must be handled separately.
Example: Determine Whether Lines Are Perpendicular
Compare
\[ y=4x+7 \]and
\[ y=-\frac14x-2. \]The slope product is
\[ 4\left(-\frac14\right) = -1. \]Therefore, the lines are
\[ \boxed{\text{perpendicular}}. \]Not Every Opposite-Sign Pair Is Perpendicular
Compare slopes
\[ 2 \]and
\[ -3. \]The signs are opposite, but
\[ 2(-3)=-6\ne-1. \]Therefore, the lines are not perpendicular.
Reciprocal Without Sign Change Is Not Perpendicular
Compare slopes
\[ \frac23 \]and
\[ \frac32. \]They are reciprocals, but
\[ \frac23\cdot\frac32=1. \]Therefore, the lines are not perpendicular.
Constructing a Parallel Line Through a Point
Suppose we want a line parallel to
\[ y=3x-4 \]through the point
\[ (2,7). \]A parallel line must have the same slope:
\[ m=3. \]Use point-slope form:
\[ y-7=3(x-2). \]Therefore,
\[ \boxed{ y-7=3(x-2) }. \]In slope-intercept form:
\[ \boxed{ y=3x+1 }. \]Constructing a Perpendicular Line Through a Point
Suppose we want a line perpendicular to
\[ y=3x-4 \]through
\[ (2,7). \]The original slope is
\[ 3. \]The perpendicular slope is
\[ -\frac13. \]Use point-slope form:
\[ \boxed{ y-7=-\frac13(x-2) }. \]Parallel Line to a Standard-Form Equation
Find a line parallel to
\[ 2x+5y=10 \]through
\[ (3,-1). \]The original line has slope
\[ m=-\frac25. \]A parallel line has the same slope.
Using point-slope form:
\[ y+1=-\frac25(x-3). \]Multiply by \(5\):
\[ 5y+5=-2x+6. \]Rearrange:
\[ \boxed{ 2x+5y=1 }. \]A Shortcut for Parallel Standard-Form Lines
If
\[ Ax+By=C \]describes one line, then any distinct parallel line can be written
\[ \boxed{ Ax+By=D }, \]with
\[ D\ne C. \]The coefficients \(A\) and \(B\) preserve the line's direction.
Only the constant changes its position.
Example of the Standard-Form Shortcut
A line parallel to
\[ 3x-4y=12 \]through
\[ (2,5) \]can be written
\[ 3x-4y=D. \]Substitute the point:
\[ 3(2)-4(5)=D. \]So
\[ D=6-20=-14. \]Therefore,
\[ \boxed{ 3x-4y=-14 }. \]Perpendicular Lines from Standard Form
For
\[ Ax+By=C, \]a direction vector is
\[ (B,-A), \]while a normal vector is
\[ (A,B). \]A perpendicular line can use the original line's normal vector as its direction vector.
This creates a useful coefficient relationship.
A Standard-Form Perpendicular Pattern
If one line is
\[ Ax+By=C, \]then a perpendicular line may be written in the form
\[ \boxed{ Bx-Ay=D } \]for some constant \(D\).
When the slopes are finite and nonzero, the slope of the first line is
\[ -\frac{A}{B}, \]while the slope of the second is
\[ \frac{B}{A}. \]The product is
\[ -\frac{A}{B}\cdot\frac{B}{A} = -1. \]Example: Perpendicular Line in Standard Form
Find a line perpendicular to
\[ 2x+3y=7 \]through
\[ (4,1). \]A perpendicular coefficient pattern is
\[ 3x-2y=D. \]Substitute the point:
\[ 3(4)-2(1)=D. \]So
\[ D=10. \]Therefore,
\[ \boxed{ 3x-2y=10 }. \]Check the Perpendicular Example by Slopes
The original line
\[ 2x+3y=7 \]has slope
\[ -\frac23. \]The new line
\[ 3x-2y=10 \]has slope
\[ \frac32. \]The product is
\[ -\frac23\cdot\frac32 = -1. \]The lines are perpendicular.
Vertical Line Through a Point
The vertical line through
\[ (5,-2) \]is simply
\[ \boxed{x=5}. \]The \(y\)-coordinate of the point does not affect the vertical line's equation because every point on the line has the same \(x\)-coordinate.
Horizontal Line Through a Point
The horizontal line through
\[ (5,-2) \]is
\[ \boxed{y=-2}. \]The \(x\)-coordinate does not affect the horizontal line's equation because every point on the line has the same \(y\)-coordinate.
Perpendicular to a Horizontal Line
Suppose the given line is
\[ y=6. \]Any perpendicular line must be vertical.
Through the point
\[ (3,-4), \]the perpendicular line is
\[ \boxed{x=3}. \]Perpendicular to a Vertical Line
Suppose the given line is
\[ x=-5. \]Any perpendicular line must be horizontal.
Through
\[ (2,7), \]the perpendicular line is
\[ \boxed{y=7}. \]Parallel to a Vertical Line
A line parallel to
\[ x=4 \]must also be vertical.
Through the point
\[ (-3,8), \]the required line is
\[ \boxed{x=-3}. \]Parallel to a Horizontal Line
A line parallel to
\[ y=-2 \]must also be horizontal.
Through
\[ (6,5), \]the required line is
\[ \boxed{y=5}. \]Parallelism as Equal Direction
Parallel lines do not merely have similar-looking graphs.
They share the same direction vector up to nonzero scaling.
For example, slopes
\[ \frac23 \]and
\[ \frac{4}{6} \]are equal because
\[ \frac{4}{6}=\frac23. \]The direction vectors
\[ (3,2) \]and
\[ (6,4) \]are scalar multiples.
Perpendicularity as Orthogonal Direction
Perpendicular lines have direction vectors whose dot product is zero.
For example, the line with slope
\[ \frac23 \]can use direction vector
\[ (3,2). \]A perpendicular direction is
\[ (-2,3). \]The dot product is
\[ (3)(-2)+(2)(3)=0. \]The corresponding perpendicular slope is
\[ \frac{3}{-2} = -\frac32. \]Equation Families with Fixed Slope
The family
\[ y=mx+b \]with fixed \(m\) and varying \(b\) consists of parallel lines.
For example,
\[ y=2x+b \]for different values of \(b\) gives an entire family of lines with slope \(2\).
Changing \(b\) translates the line vertically without changing its direction.
Equation Families in Standard Form
Similarly,
\[ Ax+By=C \]with fixed \(A\) and \(B\) but varying \(C\) describes a family of parallel lines.
The normal vector
\[ (A,B) \]remains fixed.
Changing \(C\) shifts the line along that normal direction.
Distance Between Parallel Lines Preview
Parallel lines have a constant perpendicular separation.
For normalized standard-form lines such as
\[ Ax+By=C_1 \]and
\[ Ax+By=C_2, \]their separation depends on the difference
\[ |C_2-C_1|. \]A later geometry treatment can develop the exact distance formula.
Parallel and Perpendicular Through One Point
Given one line and a point not necessarily on it:
- there is exactly one line through the point parallel to the given line;
- there is exactly one line through the point perpendicular to the given line.
These uniqueness properties are fundamental in Euclidean geometry.
Example: Same Point, Two Different Constructions
Given
\[ y=\frac12x+4 \]and the point
\[ (6,-1), \]the parallel line has slope
\[ \frac12, \]so
\[ y+1=\frac12(x-6). \]The perpendicular slope is
\[ -2, \]so
\[ y+1=-2(x-6). \]The two constructed lines meet at the given point and are perpendicular to one another only if their slopes are negative reciprocals—which they are here.
Determining Relationships from Two Equations
Suppose we are given
\[ 2x-3y=8 \]and
\[ 3x+2y=5. \]The first slope is
\[ m_1=\frac23. \]The second slope is
\[ m_2=-\frac32. \]Then
\[ m_1m_2 = \frac23\left(-\frac32\right) = -1. \]Therefore, the lines are perpendicular.
Determine Whether Lines Are Parallel, Perpendicular, or Neither
A useful classification process is:
- find or identify each line's slope;
- if the slopes are equal, check whether the lines are identical or distinct parallel lines;
- if the slopes multiply to \(-1\), the nonvertical lines are perpendicular;
- handle horizontal and vertical pairs separately;
- otherwise, classify the lines as neither.
Example: Neither
Compare
\[ y=2x+1 \]and
\[ y=-x+4. \]The slopes are
\[ 2 \]and
\[ -1. \]They are not equal.
Their product is
\[ -2\ne-1. \]So the lines are
\[ \boxed{\text{neither parallel nor perpendicular}}. \]Parallel and Perpendicular Lines in Coordinate Geometry
These relationships appear throughout geometry.
Examples include:
- opposite sides of rectangles and parallelograms;
- coordinate axes;
- altitudes of triangles;
- tangent and normal lines;
- grid systems;
- construction layouts;
- engineering drawings.
Example: Rectangle Sides
Suppose one side of a rectangle lies on a line with slope
\[ \frac34. \]The opposite side must have slope
\[ \boxed{\frac34}. \]The adjacent sides must have perpendicular slope
\[ \boxed{-\frac43}. \]Example: Triangle Altitude
Suppose one side of a triangle lies on a line with slope
\[ -2. \]An altitude drawn to that side must be perpendicular to it.
Its slope is therefore
\[ \boxed{\frac12}. \]If the opposite vertex is known, point-slope form gives the altitude equation immediately.
Example: Find an Altitude Equation
Suppose a triangle side lies on
\[ y=-2x+7 \]and the opposite vertex is
\[ (3,4). \]The side slope is
\[ -2. \]The perpendicular slope is
\[ \frac12. \]So the altitude through \((3,4)\) is
\[ \boxed{ y-4= \frac12(x-3) }. \]Example: Road Design
Suppose one straight road follows
\[ y=\frac25x+6. \]A second road is planned to cross it at a right angle.
The second road must have slope
\[ -\frac52. \]If it must pass through
\[ (10,3), \]its equation is
\[ \boxed{ y-3=-\frac52(x-10) }. \]Example: Parallel Track
A rail line follows
\[ 3x-4y=12. \]A parallel track through
\[ (8,1) \]can keep the same left-side coefficients:
\[ 3x-4y=D. \]Substitute:
\[ 3(8)-4(1)=D, \] \[ D=20. \]So the parallel track is
\[ \boxed{ 3x-4y=20 }. \]Coordinate Scale and Perpendicular Appearance
Perpendicularity is a geometric property, but a graph can look distorted if the horizontal and vertical axes use different physical scales.
Two mathematically perpendicular lines may not appear to meet at a visual \(90^\circ\) angle on a stretched graph.
Use coordinate relationships, slope, or vectors rather than appearance alone.
Parallel Appearance Can Also Be Misleading
Unequal axis scaling changes the visual angle of lines.
However, if two lines have equal numerical slopes in the same coordinate system, they remain parallel regardless of how the display is stretched.
Algebraic structure is more reliable than visual estimation.
A Reliable Horizontal-or-Vertical Workflow
- If the equation is \(y=c\), identify a horizontal line.
- Assign slope \(0\).
- If the equation is \(x=c\), identify a vertical line.
- State that its slope is undefined.
- Do not force a vertical line into \(y=mx+b\).
A Reliable Parallel-Line Workflow
- Determine the slope or direction of the given line.
- Use the same slope for a nonvertical parallel line.
- For a vertical line, construct another vertical line.
- Use the required point to determine position.
- Check that the new line is distinct if the problem asks for a parallel line rather than the same line.
A Reliable Perpendicular-Line Workflow
- Determine the slope of the original line.
- If the slope is finite and nonzero, take its negative reciprocal.
- If the original is horizontal, use a vertical perpendicular.
- If the original is vertical, use a horizontal perpendicular.
- Use the required point to construct the new line.
- Verify the slope product is \(-1\) when both slopes are finite.
A Reliable Classification Workflow
- Rewrite equations in a form that reveals direction or slope.
- Check for horizontal or vertical special cases.
- Compare slopes.
- Equal slopes indicate parallel or identical lines.
- A slope product of \(-1\) indicates perpendicular nonvertical lines.
- Otherwise classify the pair as neither.
Common Errors
Calling a Vertical Line's Slope Zero
Horizontal lines have slope \(0\). Vertical lines have undefined slope.
Calling Equal-Slope Equations Automatically Parallel
They may represent the same line. Check whether the equations are equivalent.
Changing Only the Sign for a Perpendicular Slope
The perpendicular slope is the negative reciprocal, not merely the negative.
Taking Only the Reciprocal
The sign must also change.
Applying the Negative-Reciprocal Rule to Slope Zero
A horizontal line's perpendicular is vertical, so the ordinary reciprocal expression is undefined.
Trying to Assign a Finite Slope to a Vertical Line
No real finite slope exists for a vertical line.
Ignoring Standard-Form Coefficient Scaling
Proportional \(A\) and \(B\) coefficients indicate the same direction even if the equations are scaled differently.
Trusting the Drawing More Than the Algebra
Axis scaling can distort visual angles. Use slopes or vectors to verify relationships.
Practice
1. Horizontal or Vertical?
Classify
\[ y=-8. \]Check Answer
The line is horizontal.
Its slope is
\[ \boxed{0}. \]2. Vertical Line
Classify
\[ x=6. \]Check Answer
The line is vertical.
Its slope is
\[ \boxed{\text{undefined}}. \]3. Parallel or Not?
Compare
\[ y=5x+2 \]and
\[ y=5x-9. \]Check Answer
Both have slope
\[ 5. \]The intercepts differ, so the lines are distinct and
\[ \boxed{\text{parallel}}. \]4. Same Line or Parallel?
Compare
\[ 2x+4y=8 \]and
\[ x+2y=4. \]Check Answer
The first equation divided by \(2\) becomes
\[ x+2y=4. \]Therefore, they represent
\[ \boxed{\text{the same line}}. \]5. Perpendicular Slopes
Find the slope perpendicular to
\[ m=\frac37. \]Check Answer
Take the reciprocal and change the sign:
\[ \boxed{-\frac73}. \]6. Negative Slope
Find the slope perpendicular to
\[ m=-5. \]Check Answer
\[ \boxed{\frac15}. \]Check:
\[ -5\left(\frac15\right)=-1. \]7. Parallel Through a Point
Find a line parallel to
\[ y=-2x+3 \]through
\[ (4,1). \]Check Answer
The parallel slope is
\[ -2. \]Point-slope form:
\[ y-1=-2(x-4). \]So
\[ \boxed{ y=-2x+9 }. \]8. Perpendicular Through a Point
Find a line perpendicular to
\[ y=\frac14x-7 \]through
\[ (2,5). \]Check Answer
The perpendicular slope is
\[ -4. \]So
\[ y-5=-4(x-2). \]Therefore,
\[ \boxed{ y=-4x+13 }. \]9. Perpendicular to a Vertical Line
Find the line through
\[ (3,-6) \]perpendicular to
\[ x=8. \]Check Answer
A line perpendicular to a vertical line is horizontal.
Through \((3,-6)\), it is
\[ \boxed{y=-6}. \]10. Parallel Standard Form
Find a line parallel to
\[ 4x+3y=12 \]through
\[ (2,-5). \]Check Answer
Keep the same left-side coefficients:
\[ 4x+3y=D. \]Substitute:
\[ 4(2)+3(-5)=D, \] \[ 8-15=D, \] \[ D=-7. \]Therefore,
\[ \boxed{ 4x+3y=-7 }. \]11. Perpendicular Standard Form
Find a line perpendicular to
\[ 2x-5y=9 \]through
\[ (1,4). \]Check Answer
A perpendicular coefficient pattern is
\[ -5x-2y=D. \]Multiplying by \(-1\) gives the equivalent pattern
\[ 5x+2y=D. \]Substitute \((1,4)\):
\[ 5(1)+2(4)=D, \] \[ D=13. \]Therefore,
\[ \boxed{ 5x+2y=13 }. \]12. Classify the Relationship
Compare
\[ 3x-2y=7 \]and
\[ 2x+3y=4. \]Check Answer
The first slope is
\[ \frac32. \]The second slope is
\[ -\frac23. \]The product is
\[ \frac32\left(-\frac23\right)=-1. \]Therefore, the lines are
\[ \boxed{\text{perpendicular}}. \]Key Ideas
- A horizontal line has equation \(y=c\) and slope \(0\).
- A vertical line has equation \(x=c\) and undefined slope.
- Horizontal lines are constant functions; vertical lines are not functions of \(x\).
- Distinct nonvertical parallel lines have equal slopes.
- Distinct vertical lines are parallel even though their slopes are undefined.
- Equal slope does not distinguish parallel lines from identical lines; the equations must also be checked for equivalence.
- For finite nonzero slopes, perpendicular lines satisfy \(m_1m_2=-1\).
- The perpendicular slope is the negative reciprocal: \(m_\perp=-1/m\).
- A horizontal line is perpendicular to every vertical line.
- Parallel lines through a given point preserve the original direction.
- Perpendicular lines through a given point use the orthogonal direction.
- In standard form, fixed coefficients \(A\) and \(B\) with varying \(C\) generate a family of parallel lines.
- The vector \((A,B)\) is normal to \(Ax+By=C\), providing another way to understand perpendicularity.
- Parallelism and perpendicularity can be verified using slopes, direction vectors, normal vectors, or standard-form coefficients.
- Axis scaling can distort visual appearance, so algebraic verification is more reliable than judging a graph by eye.
Where This Leads
Horizontal, vertical, parallel, and perpendicular lines turn slope into a geometric classification tool. The same rate information used to describe change now identifies direction, repeated direction, and right-angle relationships across coordinate geometry.
From here, these ideas can be applied to graphing lines efficiently, constructing equations from geometric conditions, identifying shapes from coordinates, building systems of linear equations, analyzing constraints, and later studying vectors, distances, projections, tangent and normal lines, and higher-dimensional geometry.