Training Lines and Systems of Linear Equations Applications of Systems
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Applications of Systems

22 min Lines and Systems of Linear Equations

Applications of Systems of Equations

Systems of equations are not just an algebra exercise — they model real situations where two conditions must be satisfied at the same time. Mixture problems, break-even analysis, and motion problems are classic examples.

This lesson focuses on translating word problems into systems of two equations. The hardest part is setting up the equations correctly; once you have them, you can solve by substitution or elimination.

With practice, you will develop an intuition for recognizing when a problem requires a system rather than a single equation.

Example 1 — Mixture

A chemist needs 10 L of 40% acid. She has 20% and 60% solutions. How much of each?

$x + y = 10$ and $0.20x + 0.60y = 4$.

From eq 1: $x = 10 - y$. Sub: $0.20(10 - y) + 0.60y = 4$.

$2 + 0.40y = 4 \;\Rightarrow\; y = 5$, $x = 5$. 5 L each.

Example 2 — Distance-Rate-Time

Two trains leave the same station opposite directions at 60 mph and 80 mph. When are they 420 mi apart?

$60t + 80t = 420 \;\Rightarrow\; 140t = 420 \;\Rightarrow\; t = 3$ hours.

Example 3 — Cost

Adult tickets $\$12$, child $\$8$. 350 tickets sold for $\$3{,}400$. How many of each?

$a + c = 350$ and $12a + 8c = 3400$.

$a = 350 - c$; $12(350 - c) + 8c = 3400 \;\Rightarrow\; -4c = -800 \;\Rightarrow\; c = 200$, $a = 150$.

Example 4 — Coins

A jar has nickels and dimes totaling $\$4.50$ with 60 coins. How many of each?

$n + d = 60$ and $0.05n + 0.10d = 4.50$.

$n = 60 - d$; $0.05(60 - d) + 0.10d = 4.50 \;\Rightarrow\; 3 + 0.05d = 4.50 \;\Rightarrow\; d = 30$, $n = 30$.

Practice Problems

1. Coffee A: $\$8$/lb, Coffee B: $\$14$/lb. Blend 20 lbs at $\$10$/lb. How much of each?
2. Two cars start 400 mi apart, drive toward each other at 50 and 30 mph. Meet time?
3. Quarters and dimes, 30 coins totaling $\$5.40$. How many of each?
4. Boat goes 24 mi upstream in 3 hr, 24 mi downstream in 2 hr. Boat speed and current?
5. Two numbers sum to 45. One is 3 more than twice the other.
6. Mix 30% and 70% alcohol to get 20 L of 55%. How much of each?
7. 100 tickets: student $\$5$, general $\$8$. Revenue $\$680$. How many of each?
8. Plane flies 600 mi with wind in 2 hr and against wind in 3 hr. Plane and wind speeds?
Show Answer Key

1. $\dfrac{4}{3} \times 20$: A $=$ $13\tfrac{1}{3}$ lb, B $=$ $6\tfrac{2}{3}$ lb

2. $5$ hr

3. $16$ quarters, $14$ dimes

4. Boat $= 10$ mph, current $= 2$ mph

5. $14$ and $31$

6. $7.5$ L of 30%, $12.5$ L of 70%

7. $40$ student, $60$ general

8. Plane $= 250$ mph, wind $= 50$ mph

Solving Systems by Elimination Back to Module