Algebra 1 · Unit 1
Decide whether a linear equation in one variable has one solution, no solution, or infinitely many solutions.
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One solution
After you simplify, you get x = a number. That number is the only solution.
How many solutions? 4x + 7 = 19
4x + 7 = 19
Subtract 7 from both sides.
4x = 12
Divide both sides by 4.
x = 3
You get x equal to one number, so there is one solution.
Answer: one solution, x = 3
No solution
The variable cancels and you are left with a false number sentence such as 5 = 8. That is a MATH LIE!! No value of x works.
How many solutions? 2x + 5 = 2x + 1
2x + 5 = 2x + 1
Subtract 2x from both sides.
5 = 1
5 = 1 is a MATH LIE!!
No value of x works.
Answer: no solution
Infinitely many solutions
The variable cancels and you are left with a true number sentence such as 5 = 5. That is a MATH FACT!! Every real number works. The two sides are the same expression.
How many solutions? 3x + 6 = 3x + 6
3x + 6 = 3x + 6
Subtract 3x from both sides.
6 = 6
6 = 6 is a MATH FACT!!
Every real number works.
Answer: infinitely many solutions
Special case: x = 0
If you get x = 0, the equation still has one solution. Zero is a number. Students sometimes think zero cannot be a solution, but it can.
How many solutions? 5x + 4 = 4
5x + 4 = 4
Subtract 4 from both sides.
5x = 0
Divide both sides by 5.
x = 0
Zero is a real number, so it counts. Zero can be a solution.
Answer: one solution, x = 0
The variable is already collected. Read the last line.
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
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
6x − 2 = 10
6x = 12
x = 2
One solution.
Answer: one solution
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
2(x + 4) = 2x + 8
2x + 8 = 2x + 8
8 = 8
8 = 8 is a MATH FACT!!
Infinitely many solutions.
Answer: infinitely many solutions
3(x − 1) = 3x + 2
3x − 3 = 3x + 2
−3 = 2
−3 = 2 is a MATH LIE!!
No solution.
Answer: no solution
5x − 4 = 2x + 5
Subtract 2x from both sides.
3x − 4 = 5
3x = 9
x = 3
One solution.
Answer: one solution
2x + 6 − x = x + 6
x + 6 = x + 6
6 = 6
6 = 6 is a MATH FACT!!
Infinitely many solutions.
Answer: infinitely many solutions
7 − 3x = 2 − 3x
Add 3x to both sides.
7 = 2
7 = 2 is a MATH LIE!!
No solution.
Answer: no solution
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
0.5(2x + 8) = x + 5
x + 4 = x + 5
4 = 5
4 = 5 is a MATH LIE!!
No solution.
Answer: no solution
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
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
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
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
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
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
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
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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