Linear · Systems of equations · 4.4

Solve by elimination.

Elimination: add or subtract the equations so one variable cancels, leaving one equation to solve. Then back-substitute. Lining up y = x + 1 and y = −x + 5 and subtracting eliminates y.

01 / THE IDEA

See it work.

Watch the method

Solve by elimination

Combine the equations to cancel a variable.

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{y=x+1y=−x+5\begin{cases} y = x + 1 \\ y = -x + 5 \end{cases}
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1

The system

Two lines — the solution is the point on both.

y=x+1;  y=−x+5y = x + 1;\; y = -x + 5
2

Add/subtract to eliminate a variable

Solve for x, then back-substitute for y.

x=2x = 2
3

The solution

Write the intersection point.

(2,3)(2, 3)
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.

Cancel a variable

Arrange so adding (or subtracting) removes one variable, then solve the rest.

Why it works

Knock out y, find x.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

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

1

The system

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

2

Add/subtract to eliminate a variable

Solve for x, then back-substitute for y. x = -3

3

The solution

Write the intersection point. (-3, -1)

EX 2medium

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

1

The system

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

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)

EX 3hard

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

1

The system

Two lines — the solution is the point on both. y = -x - 1;\; y = 2x + 5

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, 1)

04 / COMMON TRAPS

Where students slip.

Trap

Adding when you should subtract.

Fix

Choose the operation that cancels a variable.

Trap

Forgetting the second variable.

Fix

Back-substitute to finish.

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-6 and y=2x-10. Give the point (x, y).

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
C2

Solve the system: y=2x-3 and y=-2x+5. Give the point (x, y).

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
C3

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

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
C4

Solve the system: y=2x-3 and y=3x-7. Give the point (x, y).

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
Apply

Apply (4)

A1

Solve the system: y=x-5 and y=-2x+7. Give the point (x, y).

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
A2

Solve the system: y=-2x-5 and y=-x-3. Give the point (x, y).

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
A3

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

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
A4

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

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
Extension

Extension (4)

E1

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

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
E2

Solve the system: y=2x-9 and y=-x+3. Give the point (x, y).

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
E3

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

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.
E4

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

Need a hint?
  • Combine the equations to cancel a variable.
  • Solve, then back-substitute.