Question

8.ee.7
solving linear equations with infinite and no solution practice
solve the following equations. some equations will have a single answer, others will have no solution, and still others will have infinite solutions.

1. $2x + 2x + 2 = 4x + 2$
2. $3(x - 1) = 2x + 9$
3. $2x + 8 = 2(x + 4)$

1. $2x - x + 7 = x + 3 + 4$
2. $-2(x + 1) = -2x + 5$
3. $4x + 2x + 2 = 3x - 7$

1. $2(x + 2) + 3x = 2(x + 1) + 1$
2. $4(x - 1) = \frac{1}{2}(x - 8)$
3. $x + 2x + 7 = 3x - 7$

Explanation:

1. Equation 1: Combine like terms

Left side: $2x + 2x + 2 = 4x + 2$
Right side: $4x + 2$

1. Equation 1: Compare both sides

$4x + 2 = 4x + 2$
Subtract $4x + 2$ from both sides: $0=0$

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2. Equation 2: Expand left side

$3(x-1) = 3x - 3$

2. Equation 2: Isolate $x$

$3x - 3 = 2x + 9$
Subtract $2x$ from both sides: $x - 3 = 9$
Add 3 to both sides: $x = 9 + 3$

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3. Equation 3: Expand right side

$2(x+4) = 2x + 8$

3. Equation 3: Compare both sides

$2x + 8 = 2x + 8$
Subtract $2x + 8$ from both sides: $0=0$

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4. Equation 4: Combine like terms

Left side: $2x - x + 7 = x + 7$
Right side: $x + 3 + 4 = x + 7$

4. Equation 4: Compare both sides

$x + 7 = x + 7$
Subtract $x + 7$ from both sides: $0=0$

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5. Equation 5: Expand left side

$-2(x+1) = -2x - 2$

5. Equation 5: Compare both sides

$-2x - 2 = -2x + 5$
Add $2x$ to both sides: $-2 = 5$

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6. Equation 6: Combine like terms

Left side: $4x + 2x + 2 = 6x + 2$

6. Equation 6: Isolate $x$

$6x + 2 = 3x - 7$
Subtract $3x$ from both sides: $3x + 2 = -7$
Subtract 2 from both sides: $3x = -7 - 2$
Divide by 3: $x = \frac{-9}{3}$

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7. Equation 7: Expand and combine terms

Left side: $2(x+2)+3x = 2x + 4 + 3x = 5x + 4$
Right side: $2(x+1)+1 = 2x + 2 + 1 = 2x + 3$

7. Equation 7: Isolate $x$

$5x + 4 = 2x + 3$
Subtract $2x$ from both sides: $3x + 4 = 3$
Subtract 4 from both sides: $3x = 3 - 4$
Divide by 3: $x = \frac{-1}{3}$

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8. Equation 8: Expand both sides

Left side: $4(x-1) = 4x - 4$
Right side: $\frac{1}{2}(x-8) = \frac{1}{2}x - 4$

8. Equation 8: Isolate $x$

$4x - 4 = \frac{1}{2}x - 4$
Add 4 to both sides: $4x = \frac{1}{2}x$
Subtract $\frac{1}{2}x$ from both sides: $\frac{8}{2}x - \frac{1}{2}x = 0$
Simplify: $\frac{7}{2}x = 0$
Multiply by $\frac{2}{7}$: $x = 0$

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9. Equation 9: Combine like terms

Left side: $x + 2x + 7 = 3x + 7$

9. Equation 9: Compare both sides

$3x + 7 = 3x - 7$
Subtract $3x$ from both sides: $7 = -7$

Answer:

1. Infinite solutions
2. $x = 12$
3. Infinite solutions
4. Infinite solutions
5. No solution
6. $x = -3$
7. $x = -\frac{1}{3}$
8. $x = 0$
9. No solution