Linear · Apply · 5.2

Mixture & money.

Many real problems — coins, tickets at two prices, mixtures — are systems. Write a “count” equation and a “value” equation, then solve.

01 / THE IDEA

See it work.

Watch the method

Mixture & money

Count equation + value equation → solve.

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{y=2x−1y=x+2\begin{cases} y = 2x - 1 \\ y = x + 2 \end{cases}
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1

The system

Two lines — the solution is the point on both.

y=2x−1;  y=x+2y = 2x - 1;\; y = x + 2
2

Add/subtract to eliminate a variable

Solve for x, then back-substitute for y.

x=3x = 3
3

The solution

Write the intersection point.

(3,5)(3, 5)
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type (x, y).

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Answer:
— / WHY IT WORKS

The idea underneath.

Two relationships

One equation for the totals/counts, one for the values/amounts; solve together.

Why it works

How many, and how much.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Solve the system: y=2x+7 and y=-x+4. Give the point (x, y).

1

The system

Two lines — the solution is the point on both. y = 2x + 7;\; y = -x + 4

2

Add/subtract to eliminate a variable

Solve for x, then back-substitute for y. x = -1

3

The solution

Write the intersection point. (-1, 5)

EX 2medium

Solve the system: y=x+1 and y=2x-1. Give the point (x, y).

1

The system

Two lines — the solution is the point on both. y = x + 1;\; y = 2x - 1

2

Add/subtract to eliminate a variable

Solve for x, then back-substitute for y. x = 2

3

The solution

Write the intersection point. (2, 3)

EX 3hard

Solve the system: y=-2x+10 and y=2x-6. Give the point (x, y).

1

The system

Two lines — the solution is the point on both. y = -2x + 10;\; y = 2x - 6

2

Add/subtract to eliminate a variable

Solve for x, then back-substitute for y. x = 4

3

The solution

Write the intersection point. (4, 2)

04 / COMMON TRAPS

Where students slip.

Trap

Using one equation for everything.

Fix

Separate the count from the value.

Trap

Unit slips.

Fix

Keep the units consistent.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Solve the system: y=-x-2 and y=2x-5. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
C2

Solve the system: y=x-1 and y=2x-5. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
C3

Solve the system: y=2x-6 and y=x-2. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
C4

Solve the system: y=-2x+1 and y=-x+3. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
Apply

Apply (4)

A1

Solve the system: y=x+3 and y=-2x-3. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
A2

Solve the system: y=-2x+0 and y=-3x+1. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
A3

Solve the system: y=2x-8 and y=3x-11. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
A4

Solve the system: y=x+2 and y=2x+0. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
Extension

Extension (4)

E1

Solve the system: y=2x-3 and y=x-2. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
E2

Solve the system: y=2x+0 and y=-x-3. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
E3

Solve the system: y=-2x-1 and y=x+8. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.
E4

Solve the system: y=2x+11 and y=-x+2. Give the point (x, y).

Need a hint?
  • Write a count equation and a value equation.
  • Solve the system.