SAT · Algebra
Find the constant that makes the equation have no solution.
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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
3(x − 1) = 3x + 2
3x − 3 = 3x + 2
−3 = 2
−3 = 2 is a MATH LIE!!
No solution.
Answer: no solution
7 − 3x = 2 − 3x
Add 3x to both sides.
7 = 2
7 = 2 is a MATH LIE!!
No solution.
Answer: no solution
Distribute first, then decide.
0.5(2x + 8) = x + 5
x + 4 = x + 5
4 = 5
4 = 5 is a MATH LIE!!
No solution.
Answer: no solution
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
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
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
The two sides look different until you simplify.
4(tx + 9) = 207x + 40
Distribute.
4tx + 36 = 207x + 40
For no solutions, the x coefficients match and the constants do not.
4t = 207
Divide both sides by 4.
t = 2028
t = 57
36 ≠ 40, so the constants do not match.
Answer: t = 57
5(vx + 8) = 356x + 45
Distribute.
5vx + 40 = 356x + 45
For no solutions, the x coefficients match and the constants do not.
5v = 356
Divide both sides by 5.
v = 3530
v = 76
40 ≠ 45, so the constants do not match.
Answer: v = 76
6(yx + 11) = 245x + 72
Distribute.
6yx + 66 = 245x + 72
For no solutions, the x coefficients match and the constants do not.
6y = 245
Divide both sides by 6.
y = 2430
y = 45
66 ≠ 72, so the constants do not match.
Answer: y = 45
4(rx − d) = −3611x − 207
Distribute.
4rx − 4d = −3611x − 207
For no solutions, the x coefficients match and the constants do not.
4r = −3611
Divide both sides by 4.
r = −3644
r = −911
Answer: r = −911
5(wx − f) = −458x − 143
Distribute.
5wx − 5f = −458x − 143
For no solutions, the x coefficients match and the constants do not.
5w = −458
Divide both sides by 5.
w = −4540
w = −98
Answer: w = −98
3(qx − g) = −2110x − 256
Distribute.
3qx − 3g = −2110x − 256
For no solutions, the x coefficients match and the constants do not.
3q = −2110
Divide both sides by 3.
q = −2130
q = −710
Answer: q = −710
−5x + 20cx = 60
Factor x.
x(−5 + 20c) = 60
For no solutions, the coefficient of x is 0 and the remaining sentence is false.
−5 + 20c = 0
20c = 5
c = 520
c = 14
0 = 60 is false, so there is no solution.
Answer: c = 14
−8x + 28hx = 56
Factor x.
x(−8 + 28h) = 56
For no solutions, the coefficient of x is 0 and the remaining sentence is false.
−8 + 28h = 0
28h = 8
h = 828
h = 27
0 = 56 is false, so there is no solution.
Answer: h = 27
−6x + 15sx = 45
Factor x.
x(−6 + 15s) = 45
For no solutions, the coefficient of x is 0 and the remaining sentence is false.
−6 + 15s = 0
15s = 6
s = 615
s = 25
0 = 45 is false, so there is no solution.
Answer: s = 25
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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