Linear · Systems of equations · 4.5

How many solutions?

Compare the lines: different slopes cross once (one solution); the same slope but different intercepts are parallel (no solution); the same line twice has infinitely many solutions.

01 / THE IDEA

See it work.

Watch the method

How many solutions?

Different slopes → one; parallel → none; same line → infinite.

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

Compare the slopes

Different slopes → one solution; same slope & intercept → infinite; same slope, different intercept → none.

None\text{None}
— / WHY IT WORKS

The idea underneath.

Compare the lines

Intersecting → one; parallel (no meeting) → none; identical → infinite.

Why it works

Parallel lines never cross.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

How many solutions does this system have? y=2x-3 and y=2x-3.

1

Answer

Infinitely many

EX 2medium

How many solutions does this system have? y=x+1 and y=x+1.

1

Answer

Infinitely many

EX 3hard

How many solutions does this system have? y=x-4 and y=x-3.

1

Answer

None

04 / COMMON TRAPS

Where students slip.

Trap

Assuming a system always has one solution.

Fix

Parallel or identical lines are special cases.

Trap

Confusing none and infinite.

Fix

Same slope, different intercept → none; identical → infinite.

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

How many solutions does this system have? y=-x+2 and y=-x+5.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
C2

How many solutions does this system have? y=3x+4 and y=-3x+4.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
C3

How many solutions does this system have? y=2x+4 and y=2x+7.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
C4

How many solutions does this system have? y=-2x+3 and y=3x+3.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
Apply

Apply (4)

A1

How many solutions does this system have? y=-x+1 and y=-x+4.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
A2

How many solutions does this system have? y=2x+1 and y=2x+3.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
A3

How many solutions does this system have? y=-x+4 and y=-3x+4.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
A4

How many solutions does this system have? y=x-1 and y=x+1.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
Extension

Extension (4)

E1

How many solutions does this system have? y=3x-3 and y=3x-2.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
E2

How many solutions does this system have? y=3x-1 and y=3x-1.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
E3

How many solutions does this system have? y=2x-3 and y=2x-2.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.
E4

How many solutions does this system have? y=3x+1 and y=x+1.

Need a hint?
  • Compare the slopes.
  • Then compare the intercepts.