Solution Showdown Worksheet
SOLUTION SHOWDOWN
8th Grade Mathematics • Linear Equations with Variables on Both Sides
Page 1 of 2
Name:
Date:
Period:
Directions: Show all algebraic steps in the workspace. Then, determine if the equation yields One Solution (find \(x\)), No Solution, or Infinitely Many Solutions.
1 \( 4x + 3x - 5 = 2x + 15 \)
Combining Like Terms
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
2 \( 3(2x - 4) = 6x + 5 \)
Distributive Property
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
3 \( 2(4x + 3) = 5x + 3x + 6 \)
Distribute & Combine
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
4 \( 9x - 7 - 2x = 3x + 5 + x \)
Combine Both Sides
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
5 \( 4(2x - 3) = 2(4x + 1) \)
Distribute Both Sides
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
SOLUTION SHOWDOWN (CONTINUED)
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Page 2 of 2
6 \( \frac{1}{3}(6x - 9) = x + 4 \)
Basic Fraction
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
7 \( \frac{1}{2}(8x - 14) = 6x - 2x - 7 \)
Basic Fraction
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
8 \( 3(x + 4) + 2x = 5(x + 2) + 2 \)
Distribute & Combine Both
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
9 \( 2(5x - 1) - 4x = 6x + 9 \)
Multi-Step Expansion
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
10 \( -2(3x - 5) = 4(x + 5) \)
Negative Distribution
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One Solution: \( x = \) ______ No Solution Infinitely Many Solutions
Solution Showdown Answer Key
SOLUTION SHOWDOWN
Answer Key
Teacher Edition • Complete Step-by-Step Solutions & Mathematical Notes
Key • Page 1 of 2
One Solution: Variables remain; \(x = a\)
No Solution: Variables cancel; \(a = b\) (False)
Infinitely Many: Identical; \(a = a\) (True)
1 \( 4x + 3x - 5 = 2x + 15 \)
One Solution
1. Combine left: \( 7x - 5 = 2x + 15 \)
2. Subtract \(2x\): \( 5x - 5 = 15 \)
3. Add 5 & divide by 5: \( 5x = 20 \implies \mathbf{x = 4} \)
One Solution: \( \mathbf{x = 4} \)
No Solution
Infinitely Many Solutions
2 \( 3(2x - 4) = 6x + 5 \)
No Solution
1. Distribute left: \( 6x - 12 = 6x + 5 \)
2. Subtract \(6x\) both sides: \( -12 = 5 \)
3. Verdict: False statement / Contradiction
One Solution: \( x = \) ______
No Solution
Infinitely Many Solutions
3 \( 2(4x + 3) = 5x + 3x + 6 \)
Infinitely Many
1. Distribute left & combine right: \( 8x + 6 = 8x + 6 \)
2. Subtract \(8x\) both sides: \( 6 = 6 \)
3. Verdict: True statement / Identity
One Solution: \( x = \) ______
No Solution
Infinitely Many Solutions
4 \( 9x - 7 - 2x = 3x + 5 + x \)
One Solution
1. Combine both sides: \( 7x - 7 = 4x + 5 \)
2. Subtract \(4x\): \( 3x - 7 = 5 \)
3. Add 7 & divide by 3: \( 3x = 12 \implies \mathbf{x = 4} \)
One Solution: \( \mathbf{x = 4} \)
No Solution
Infinitely Many Solutions
5 \( 4(2x - 3) = 2(4x + 1) \)
No Solution
1. Distribute both sides: \( 8x - 12 = 8x + 2 \)
2. Subtract \(8x\) both sides: \( -12 = 2 \)
3. Verdict: False statement / Contradiction
One Solution: \( x = \) ______
No Solution
Infinitely Many Solutions
SOLUTION SHOWDOWN (ANSWER KEY • CONTINUED)
Key • Page 2 of 2
6 \( \frac{1}{3}(6x - 9) = x + 4 \)
One Solution
1. Distribute fraction \( \frac{1}{3} \): \( 2x - 3 = x + 4 \)
2. Subtract \(x\) from both sides: \( x - 3 = 4 \)
3. Add 3 to both sides: \( \mathbf{x = 7} \)
One Solution: \( \mathbf{x = 7} \)
No Solution
Infinitely Many Solutions
7 \( \frac{1}{2}(8x - 14) = 6x - 2x - 7 \)
Infinitely Many
1. Distribute \( \frac{1}{2} \) & combine right: \( 4x - 7 = 4x - 7 \)
2. Subtract \(4x\) both sides: \( -7 = -7 \)
3. Verdict: True statement / Identity
One Solution: \( x = \) ______
No Solution
Infinitely Many Solutions
8 \( 3(x + 4) + 2x = 5(x + 2) + 2 \)
Infinitely Many
1. Expand & combine left: \( 3x + 12 + 2x = 5x + 12 \)
2. Expand & combine right: \( 5x + 10 + 2 = 5x + 12 \)
3. Subtract \(5x\): \( 12 = 12 \) (Identity)
One Solution: \( x = \) ______
No Solution
Infinitely Many Solutions
9 \( 2(5x - 1) - 4x = 6x + 9 \)
No Solution
1. Distribute left: \( 10x - 2 - 4x = 6x + 9 \)
2. Combine left: \( 6x - 2 = 6x + 9 \)
3. Subtract \(6x\): \( -2 = 9 \) (Contradiction)
One Solution: \( x = \) ______
No Solution
Infinitely Many Solutions
10 \( -2(3x - 5) = 4(x + 5) \)
One Solution
1. Distribute both: \( -6x + 10 = 4x + 20 \)
2. Add \(6x\) & subtract 20: \( -10 = 10x \)
3. Divide by 10: \( \mathbf{x = -1} \)
One Solution: \( \mathbf{x = -1} \)
No Solution
Infinitely Many Solutions
