Solving Linear Equations

4

Distribute 1.25 to both terms: $5x - 12.5 = 7.5$. Add 12.5 to both sides to get $5x = 20$, then divide by 5: $x = 4$.

3.5: The 1.25 was multiplied by $4x$ but not by −10, giving $5x - 10 = 7.5$ and $x = 3.5$. Distribute to every term in the parentheses.

−1: The distribution was correct, but 12.5 was subtracted from the right side instead of added, giving $5x = -5$. Undo −12.5 by adding 12.5 to both sides.

−0.5: The 1.25 wasn't distributed to −10, and then 10 was subtracted instead of added. Distribute to every term, then use inverse operations.

-38

Distribute: $-6m - 8 = 68 - 4m$. Add $4m$ to both sides: $-2m - 8 = 68$. Add 8: $-2m = 76$. Divide by −2: $m = -38$.

38: Check the last step: $-2m = 76$ means you divide by a negative, so $m = -38$.

-30: Remember the negative sign distributes to the 8 too: $-(6m + 8) = -6m - 8$.

13

Distribute: $4q + 226 = 30q - 112$. Subtract $4q$: $226 = 26q - 112$. Add 112: $338 = 26q$. Divide by 26: $q = 13$.

4.5: The 4 was multiplied by $q$ but not by 56.5. Distribute to both terms: $4 \cdot 56.5 = 226$.

5

Distribute: $80 - 10y = 10y + 5 - 5y$. Combine like terms: $80 - 10y = 5y + 5$. Add $10y$ and subtract 5: $75 = 15y$, so $y = 5$.

15: The signs were lost when distributing, so every term became positive. Keep the subtraction: $2(-5y) = -10y$ and $5(-y) = -5y$.

7.5: The 2 was distributed only to the 40, giving $80 - 5y$. Multiply the 2 by both terms inside the parentheses.

Not here: A solution is listed. Distribute carefully, combine like terms, and you get $y = 5$.

−2

Distribute: $4y-12+19=16y+24+7$, so $4y+7=16y+31$. Then $-24=12y$, so $y=-2$.

2: The solution is negative: $-24=12y$.

$-\frac{1}{2}$: Check the distribution and combine constants before dividing by 12.

$\frac{1}{2}$: The variable terms and constants must be collected on opposite sides.

32

Distribute: $-8m+28=156-12m$. Add $12m$ and subtract 28: $4m=128$, so $m=32$.

46: Check the distribution on the left: $-4(2m-7)=-8m+28$.

−6.4: Collect variable terms on one side and constants on the other before dividing by 4.

−40.75: The constants are $156-28=128$, and the coefficient is $12m-8m=4m$.

$PR = 10$ cm and $RQ = 24$ cm

The perimeter equation is $x + 16 + 3(x - 2) = 5x$, which simplifies to $4x + 10 = 5x$, so $x = 10$. Therefore $PR = 10$ cm and $RQ = 3(10 - 2) = 24$ cm.

$-21$

Distribute −4: $\frac{4}{3}z + 6 = -\frac{2}{3}z - 36$. Add $\frac{2}{3}z$ to both sides: $2z + 6 = -36$. Subtract 6: $2z = -42$, so $z = -21$.

$\frac{3}{2}$: Only the first term in the parentheses was multiplied by −4, giving $2z + 6 = 9$.

$\frac{9}{2}$: Only the first term was multiplied, and by 4 instead of −4.

$-45$: Inverse operations were not used when moving terms across the equal sign, giving $\frac{2}{3}z = -30$.

$-\frac{62}{3}$

Multiply both sides by 10: $2(v - 6) = 5(v + 10)$, so $2v - 12 = 5v + 50$. Subtract $5v$ and add 12: $-3v = 62$, so $v = -\frac{62}{3}$.

$-\frac{16}{3}$: Only the first term in each set of parentheses was multiplied, giving $2v - 6 = 5v + 10$.

$\frac{38}{7}$: Terms were moved across the equal sign without inverse operations, giving $7v = 38$.

$\frac{11}{7}$: The denominators were added to the constants instead of multiplied, and terms were moved without inverse operations.

–24

Distribute: $3v + 6 = 2v - 18$. Subtract $2v$ and 6 from both sides: $v = -24$.

–11: Only the first term in each set of parentheses was multiplied, giving $3v + 2 = 2v - 9$.

$-\frac{12}{5}$: Terms were moved across the equal sign without inverse operations, giving $5v = -12$.

$-\frac{7}{5}$: Only the first terms were multiplied, and terms were moved without inverse operations, giving $5v = -7$.