Solving Equations · Variables on both sides · 4.3

How many solutions?

Sometimes the variable cancels. If the constants then disagree (2x + 1 = 2x + 5 → 1 = 5), there is NO solution. If they agree (2(x+1) = 2x + 2 → 2 = 2), EVERY value works — infinitely many solutions.

01 / THE IDEA

See it work.

Watch the method

No solution or infinitely many

2x + 1 = 2x + 5 → 1 = 5, impossible → no solution.

1 / 2
2x+1=2x+52x + 1 = 2x + 5
1

The equation

Solve for x — find the value that makes both sides equal.

2x+1=2x+52x + 1 = 2x + 5
2

No solution

The variable cancels and the constants disagree — no value of x works.

no solution\text{no solution}
— / WHY IT WORKS

The idea underneath.

When x disappears

If both sides reduce to a false statement, no solution; to a true statement, infinitely many.

Why it works

1 = 5 is never true; 2 = 2 is always true.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

How many solutions does 5x + 4 = 5x + 8 have?

1

Answer

No solution

EX 2medium

How many solutions does 3x + 2 = 3x + 5 have?

1

Answer

No solution

EX 3hard

How many solutions does 2x + 3 = 2x + 9 have?

1

Answer

No solution

04 / COMMON TRAPS

Where students slip.

Trap

Writing x = 0 when x cancels.

Fix

A cancelled x means none or all, not zero.

Trap

Stopping before checking the constants.

Fix

Compare the leftover constants.

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 5x + 4 = 5x + 5 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
C2

How many solutions does 4x + 4 = 4x + 5 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
C3

How many solutions does 3x + 3 = 3x + 9 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
C4

How many solutions does 4x + 1 = 4x + 5 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
Apply

Apply (4)

A1

How many solutions does 5x + 4 = 5x + 8 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
A2

How many solutions does 2x + 2 = 2x + 5 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
A3

How many solutions does 4(x + 1) = 4x + 4 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
A4

How many solutions does 2x + 4 = 2x + 7 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
Extension

Extension (4)

E1

How many solutions does 5(x + 1) = 5x + 5 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
E2

How many solutions does 2x + 3 = 2x + 7 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
E3

How many solutions does 3(x + 3) = 3x + 9 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.
E4

How many solutions does 2x + 3 = 2x + 8 have?

Need a hint?
  • Collect the x terms — do they cancel?
  • Compare the leftover constants.