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SAT · Algebra

Algebra – How Many Solutions

Determine if an equation has one solution, no solution, or infinitely many solutions.

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Question Bank Easy Medium Hard
Question ID Difficulty Topic Focus
550b352c ⭐ Easy Determine number of solutions (one / none / infinite)
5a7ab8e8 ⭐ Medium Determine number of solutions (one / none / infinite)
0393ba15 ⭐⭐⭐⭐ Hard Determine number of solutions (one / none / infinite) –
Students commonly miss this question.
25e1cfed ⭐⭐⭐ Hard Determine number of solutions (one / none / infinite)
69f609b2 ⭐⭐⭐ Hard Determine number of solutions (one / none / infinite)

Tip: Search these exact Question IDs in the College Board SAT Suite Question Bank, Algebra, skill Linear equations in one variable for full questions and rationales.

Level 1

Easy

The variable is already collected. Read the last line.

Example
How many solutions? 3x + 2 = 3x + 8

3x + 2 = 3x + 8

Subtract 3x from both sides.

2 = 8

2 = 8 is a MATH LIE!!

No value of x works.

Answer: no solution

1
How many solutions? 2x + 1 = 7
Show Answer

2x + 1 = 7

Subtract 1 from both sides.

2x = 6

x = 3

One solution.

Answer: one solution

2
How many solutions? 5x + 4 = 5x + 4
Show Answer

5x + 4 = 5x + 4

Subtract 5x from both sides.

4 = 4

4 = 4 is a MATH FACT!!

Every real number is a solution.

Answer: infinitely many solutions

3
How many solutions? x + 3 = x + 3
Show Answer

Subtract x from both sides.

3 = 3

3 = 3 is a MATH FACT!!

Answer: infinitely many solutions

4
How many solutions? 4x − 1 = 4x + 2
Show Answer

4x − 1 = 4x + 2

Subtract 4x from both sides.

−1 = 2

−1 = 2 is a MATH LIE!!

No value of x works.

Answer: no solution

5
How many solutions? 6x − 2 = 10
Show Answer

6x − 2 = 10

6x = 12

x = 2

One solution.

Answer: one solution

6
How many solutions? 2(x + 4) = 2x + 8
Show Answer

2(x + 4) = 2x + 8

2x + 8 = 2x + 8

8 = 8

8 = 8 is a MATH FACT!!

Infinitely many solutions.

Answer: infinitely many solutions

7
How many solutions? 3(x − 1) = 3x + 2
Show Answer

3(x − 1) = 3x + 2

3x − 3 = 3x + 2

−3 = 2

−3 = 2 is a MATH LIE!!

No solution.

Answer: no solution

8
How many solutions? 5x − 4 = 2x + 5
Show Answer

5x − 4 = 2x + 5

Subtract 2x from both sides.

3x − 4 = 5

3x = 9

x = 3

One solution.

Answer: one solution

9
How many solutions? 2x + 6 − x = x + 6
Show Answer

2x + 6 − x = x + 6

x + 6 = x + 6

6 = 6

6 = 6 is a MATH FACT!!

Infinitely many solutions.

Answer: infinitely many solutions

10
How many solutions? 7 − 3x = 2 − 3x
Show Answer

7 − 3x = 2 − 3x

Add 3x to both sides.

7 = 2

7 = 2 is a MATH LIE!!

No solution.

Answer: no solution

Level 2

Medium

Distribute first, then decide.

Example
How many solutions? 4(x − 3) + 2 = 4x − 10

4(x − 3) + 2 = 4x − 10

Distribute.

4x − 12 + 2 = 4x − 10

4x − 10 = 4x − 10

Subtract 4x from both sides.

−10 = −10

−10 = −10 is a MATH FACT!!

Every real number works.

Answer: infinitely many solutions

1
How many solutions? 0.5(2x + 8) = x + 5
Show Answer

0.5(2x + 8) = x + 5

x + 4 = x + 5

4 = 5

4 = 5 is a MATH LIE!!

No solution.

Answer: no solution

2
How many solutions? 3(2x − 1) − x = 5x − 3
Show Answer

3(2x − 1) − x = 5x − 3

6x − 3 − x = 5x − 3

5x − 3 = 5x − 3

Subtract 5x from both sides.

−3 = −3

−3 = −3 is a MATH FACT!!

Infinitely many solutions.

Answer: infinitely many solutions

3
How many solutions? 2(x + 5) − 3 = 2x + 6
Show Answer

2(x + 5) − 3 = 2x + 6

2x + 10 − 3 = 2x + 6

2x + 7 = 2x + 6

7 = 6

7 = 6 is a MATH LIE!!

No solution.

Answer: no solution

4
How many solutions? 5(2x − 3) − 4x = 6x − 14
Show Answer

5(2x − 3) − 4x = 6x − 14

Distribute.

10x − 15 − 4x = 6x − 14

6x − 15 = 6x − 14

Subtract 6x from both sides.

−15 = −14

−15 = −14 is a MATH LIE!!

No solution.

Answer: no solution

5
How many solutions? 23(6x − 9) − 2x = 2x − 6
Show Answer

23(6x − 9) − 2x = 2x − 6

4x − 6 − 2x = 2x − 6

2x − 6 = 2x − 6

Subtract 2x from both sides.

−6 = −6

−6 = −6 is a MATH FACT!!

Infinitely many solutions.

Answer: infinitely many solutions

Level 3

Hard

The two sides look different until you simplify.

Example
How many solutions? 12(4x − 6) = 2x − 3

12(4x − 6) = 2x − 3

2x − 3 = 2x − 3

−3 = −3

−3 = −3 is a MATH FACT!!

Infinitely many solutions.

Answer: infinitely many solutions

1
How many solutions? 24x = 48x
Show Answer

24x = 48x

Subtract 24x from both sides.

0 = 24x

Divide both sides by 24.

0 = x

x = 0

Zero is a number, so there is one solution.

Answer: one solution, x = 0

2
If k(x + 2) = 3x + 6 has exactly one solution, what must be true of k?
Show Answer

k(x + 2) = 3x + 6

Distribute.

kx + 2k = 3x + 6

Subtract 3x from both sides.

kx − 3x + 2k = 6

x(k − 3) + 2k = 6

If k = 3, then 2(3) = 6, which is 6 = 6, true for every x: infinitely many solutions.

If k ≠ 3, the x term stays, so there is exactly one solution.

The equation has exactly one solution when k ≠ 3.

Answer: k ≠ 3

3
If the equation 7x + 4 = px + 4 has exactly one solution, what is that solution?
Show Answer

7x + 4 = px + 4

Subtract 4 from both sides.

7x = px

Subtract px from both sides.

7x − px = 0

x(7 − p) = 0

If p = 7, then 0 = 0, infinitely many solutions.

If p ≠ 7, divide both sides by 7 − p.

x = 0

Zero is a number, so the one solution is x = 0.

Answer: x = 0

4
How many solutions does the equation 21x = 42x have?
Show Answer

21x = 42x

Subtract 21x from both sides.

0 = 21x

Divide both sides by 21.

0 = x

x = 0

Zero is a number, so there is one solution.

Answer: one solution, x = 0

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