Linear · Systems of equations · 4.3

Solve by substitution.

Substitution: since both equations give y, set them equal (or substitute one into the other), solve for x, then back-substitute for y. 2x − 1 = −x + 5 → x = 2 → y = 3.

01 / THE IDEA

See it work.

Watch the method

Solve by substitution

Set them equal: 2x − 1 = −x + 5 → x = 2 → y = 3.

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{y=2x−1y=−x+5\begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases}
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1

The system

Two lines — the solution is the point on both.

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

Substitute one equation into the other

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.

One variable

Replace y in one equation with the other expression, solve for x, then find y.

Why it works

Two expressions for y must be equal.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

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

1

The system

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

2

Substitute one equation into the other

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+4. Give the point (x, y).

1

The system

Two lines — the solution is the point on both. y = -2x + 6;\; y = -x + 4

2

Substitute one equation into the other

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

3

The solution

Write the intersection point. (2, 2)

EX 3hard

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

1

The system

Two lines — the solution is the point on both. y = 2x - 7;\; y = -x + 5

2

Substitute one equation into the other

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

3

The solution

Write the intersection point. (4, 1)

04 / COMMON TRAPS

Where students slip.

Trap

Forgetting to find y.

Fix

Back-substitute x to get y.

Trap

Substituting into nothing.

Fix

Replace the variable in the OTHER equation.

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-1 and y=-x+5. Give the point (x, y).

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
C2

Solve the system: y=-x+0 and y=2x+6. Give the point (x, y).

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
C3

Solve the system: y=-x+1 and y=x-7. Give the point (x, y).

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
C4

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

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
Apply

Apply (4)

A1

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

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
A2

Solve the system: y=2x+8 and y=-2x-4. Give the point (x, y).

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
A3

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

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
A4

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

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
Extension

Extension (4)

E1

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

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
E2

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

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
E3

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

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.
E4

Solve the system: y=-x+4 and y=3x-8. Give the point (x, y).

Need a hint?
  • Set the two y-expressions equal.
  • Solve for x, then find y.