Linear · Systems of equations · 4.2

Solve by graphing.

Graph both lines on the same plane; the point where they cross is the solution. Read its coordinates. For y = 2x − 1 and y = −x + 5, the lines meet at (2, 3).

01 / THE IDEA

See it work.

Watch the method

Solve by graphing

Graph both; read the intersection.

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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

Find where the lines cross

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).

streak 0
best 0
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Answer:
— / WHY IT WORKS

The idea underneath.

See it cross

Plot each line accurately; the intersection’s coordinates are the solution.

Why it works

The crossing point solves both.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

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

1

The system

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

2

Find where the lines cross

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

3

The solution

Write the intersection point. (-2, 2)

EX 2medium

Solve the system: y=2x-9 and y=-x+0. Give the point (x, y).

1

The system

Two lines — the solution is the point on both. y = 2x - 9;\; y = -x

2

Find where the lines cross

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

3

The solution

Write the intersection point. (3, -3)

EX 3hard

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

1

The system

Two lines — the solution is the point on both. y = -x + 1;\; y = x + 7

2

Find where the lines cross

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

3

The solution

Write the intersection point. (-3, 4)

04 / COMMON TRAPS

Where students slip.

Trap

Misreading the intersection.

Fix

Read both coordinates carefully.

Trap

Sloppy graphing.

Fix

Use the intercept and slope for each line.

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=2x+4 and y=x+3. Give the point (x, y).

Need a hint?
  • Graph both lines.
  • Read where they cross.
C2

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

Need a hint?
  • Graph both lines.
  • Read where they cross.
C3

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

Need a hint?
  • Graph both lines.
  • Read where they cross.
C4

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

Need a hint?
  • Graph both lines.
  • Read where they cross.
Apply

Apply (4)

A1

Solve the system: y=-2x+5 and y=2x-11. Give the point (x, y).

Need a hint?
  • Graph both lines.
  • Read where they cross.
A2

Solve the system: y=-2x+5 and y=3x+0. Give the point (x, y).

Need a hint?
  • Graph both lines.
  • Read where they cross.
A3

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

Need a hint?
  • Graph both lines.
  • Read where they cross.
A4

Solve the system: y=x-3 and y=2x-4. Give the point (x, y).

Need a hint?
  • Graph both lines.
  • Read where they cross.
Extension

Extension (4)

E1

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

Need a hint?
  • Graph both lines.
  • Read where they cross.
E2

Solve the system: y=2x+11 and y=-3x-4. Give the point (x, y).

Need a hint?
  • Graph both lines.
  • Read where they cross.
E3

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

Need a hint?
  • Graph both lines.
  • Read where they cross.
E4

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

Need a hint?
  • Graph both lines.
  • Read where they cross.