Algebra 101 - Unit 13

Lesson 78: Applications of Linear Systems

Linear systems are not just abstract equations—they model real-world situations. Applications often involve problems like mixtures, motion, cost/profit, or age problems.

Steps for Solving Word Problems

1. Read the problem carefully and identify the variables.
2. Write one equation for each relationship described.
3. Solve the system using substitution or elimination.
4. Interpret the solution in the context of the problem.
5. Check your solution to ensure it makes sense.

Examples

Example 1 — Mixture Problem:

A chemist has a 10% acid solution and a 30% acid solution. How many liters of each should be mixed to get 8 liters of a 20% solution?

Let x = liters of 10% solution, y = liters of 30% solution: x + y = 8
0.10x + 0.30y = 0.20(8)

Solve: x + y = 8 → y = 8 − x
0.10x + 0.30(8 − x) = 1.6 → 0.10x + 2.4 − 0.30x = 1.6 → −0.20x = −0.8 → x = 4 → y = 4

Example 2 — Motion Problem:

Two cars start from the same point. One travels east at 60 mph, the other north at 50 mph. How far apart are they after 2 hours?

x = distance east = 60 × 2 = 120 miles
y = distance north = 50 × 2 = 100 miles
Distance apart: d = √(x² + y²) = √(120² + 100²) = √(14400 + 10000) = √24400 ≈ 156.2 miles

Example 3 — Cost/Profit Problem:

A company sells pens for $2 and pencils for $1.50. They sell 100 items and earn $175. How many pens and pencils did they sell?

Let x = pens, y = pencils: x + y = 100
2x + 1.5y = 175
Solve: y = 100 − x → 2x + 1.5(100 − x) = 175 → 2x + 150 − 1.5x = 175 → 0.5x = 25 → x = 50 → y = 50

Practice Problems

1. A mix of nuts: $4 per lb of almonds and $2 per lb of cashews. A 10-lb mix costs $30. How many pounds of each?
2. Two trains leave the same station: one at 70 mph, one at 50 mph. After 3 hours, how far apart are they if one goes north and the other east?
3. A store sells shirts for $15 and hats for $10. They sell 40 items and make $500. How many of each were sold?
4. A tank has 100 liters of solution. How much of 5% and 20% solution should be mixed to get 100 liters of 12% solution?
5. A company produces two products. Product A costs $8, product B costs $5. They produce 500 items and spend $3400. How many of each product?