SAT · Algebra
Find the constant that makes the equation have infinitely many solutions.
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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.
The variable is already collected. Read the last line.
2x + 1 = 7
Subtract 1 from both sides.
2x = 6
x = 3
One solution.
Answer: one solution
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
Subtract x from both sides.
3 = 3
3 = 3 is a MATH FACT!!
Answer: infinitely many solutions
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
2(x + 4) = 2x + 8
2x + 8 = 2x + 8
8 = 8
8 = 8 is a MATH FACT!!
Infinitely many solutions.
Answer: infinitely many solutions
2x + 6 − x = x + 6
x + 6 = x + 6
6 = 6
6 = 6 is a MATH FACT!!
Infinitely many solutions.
Answer: infinitely many solutions
Distribute first, then decide.
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
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
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
The two sides look different until you simplify.
12(4x − 6) = 2x − 3
2x − 3 = 2x − 3
−3 = −3
−3 = −3 is a MATH FACT!!
Infinitely many solutions.
Answer: infinitely many solutions
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
5(x − 4) = mx + n
Distribute.
5x − 20 = mx + n
The two sides match for every x only when the x coefficients match and the constants match.
m = 5 and n = −20
m + n = 5 + (−20) = −15
Answer: m + n = −15
a(x − 5) = 2x − 10
Distribute.
ax − 5a = 2x − 10
The two sides match for every x only when the x coefficients match and the constants match.
a = 2
Check the constants.
−5(2) = −10
Answer: a = 2
3(x + b) = 3x + 12
Distribute.
3x + 3b = 3x + 12
The x coefficients already match.
The two sides match for every x only when the constants match.
3b = 12
b = 4
Answer: b = 4
20x + 364 − a9 = b(x − 7)
Simplify the first fraction.
5x + 9 − a9 = bx − 7b
For infinitely many solutions, the two sides must be the same expression.
The x coefficients match.
5 = b
b = 5
The constants match.
9 − a9 = −7b
Substitute b = 5.
9 − a9 = −7(5)
9 − a9 = −35
Subtract 9 from both sides.
−a9 = −44
Multiply both sides by −9.
a = 396
Answer: a = 396
30x + 186 − a5 = b(x − 4)
Simplify the first fraction.
5x + 3 − a5 = bx − 4b
For infinitely many solutions, the two sides must be the same expression.
The x coefficients match.
5 = b
b = 5
The constants match.
3 − a5 = −4b
Substitute b = 5.
3 − a5 = −4(5)
3 − a5 = −20
Subtract 3 from both sides.
−a5 = −23
Multiply both sides by −5.
a = 115
Answer: a = 115
14x + 427 − a8 = b(x − 11)
Simplify the first fraction.
2x + 6 − a8 = bx − 11b
For infinitely many solutions, the two sides must be the same expression.
The x coefficients match.
2 = b
b = 2
The constants match.
6 − a8 = −11b
Substitute b = 2.
6 − a8 = −11(2)
6 − a8 = −22
Subtract 6 from both sides.
−a8 = −28
Multiply both sides by −8.
a = 224
Answer: a = 224
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