Here are the step-by-step solutions for each equation on the worksheet.
1) $8(4a - 4) - 1 = 14 - 8a$
Step 1: Distribute the 8 into the parentheses.
$$32a - 32 - 1 = 14 - 8a$$
Step 2: Combine like terms on the left side ($-32 - 1$).
$$32a - 33 = 14 - 8a$$
Step 3: Add $8a$ to both sides to get all variables on one side.
$$40a - 33 = 14$$
Step 4: Add 33 to both sides to isolate the variable term.
$$40a = 47$$
Step 5: Divide by 40.
$$a = \frac{47}{40}$$ (or $1.175$)
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2) $4(-8x + 5) = -30x - 26$
Step 1: Distribute the 4 into the parentheses.
$$-32x + 20 = -30x - 26$$
Step 2: Add $32x$ to both sides to move the variables to the right (to keep the coefficient positive).
$$20 = 2x - 26$$
Step 3: Add 26 to both sides.
$$46 = 2x$$
Step 4: Divide by 2.
$$23 = x$$
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3) $27 = 46 + 3x - x$
Step 1: Combine like terms on the right side ($3x - x = 2x$).
$$27 = 46 + 2x$$
Step 2: Subtract 46 from both sides.
$$-19 = 2x$$
Step 3: Divide by 2.
$$x = -\frac{19}{2}$$ (or $-9.5$)
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4) $38 + 7k = 8(k + 4)$
Step 1: Distribute the 8 into the parentheses on the right side.
$$38 + 7k = 8k + 32$$
Step 2: Subtract $7k$ from both sides.
$$38 = k + 32$$
Step 3: Subtract 32 from both sides.
$$6 = k$$
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5) $-2(4 - x) = 6(x + 4) + 4x$
Step 1: Distribute the numbers into the parentheses. Be careful with the negative sign on the left.
Left side: $-2 \cdot 4 = -8$ and $-2 \cdot -x = +2x$. So, $-8 + 2x$.
Right side: $6 \cdot x = 6x$ and $6 \cdot 4 = 24$. So, $6x + 24 + 4x$.
Equation becomes:
$$-8 + 2x = 6x + 24 + 4x$$
Step 2: Combine like terms on the right side ($6x + 4x = 10x$).
$$-8 + 2x = 10x + 24$$
Step 3: Subtract $2x$ from both sides.
$$-8 = 8x + 24$$
Step 4: Subtract 24 from both sides.
$$-32 = 8x$$
Step 5: Divide by 8.
$$-4 = x$$
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6) $0.05z + 0.2 = 0.15z - 10.5$
Step 1: Subtract $0.05z$ from both sides to group the $z$ terms.
$$0.2 = 0.10z - 10.5$$
Step 2: Add 10.5 to both sides.
$$10.7 = 0.10z$$
Step 3: Divide by 0.10 (which is the same as multiplying by 10).
$$107 = z$$
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7) $4(6 + 4x) = 0$
Step 1: Distribute the 4.
$$24 + 16x = 0$$
Step 2: Subtract 24 from both sides.
$$16x = -24$$
Step 3: Divide by 16.
$$x = -\frac{24}{16}$$
Step 4: Simplify the fraction (divide top and bottom by 8).
$$x = -\frac{3}{2}$$ (or $-1.5$)
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8) $\frac{x}{8} - \frac{x}{9} = 1$
Step 1: Find a common denominator for 8 and 9, which is 72.
Multiply the entire equation by 72 to clear the fractions.
$$72 \cdot (\frac{x}{8}) - 72 \cdot (\frac{x}{9}) = 72 \cdot 1$$
Step 2: Simplify.
$$9x - 8x = 72$$
Step 3: Subtract the $x$ terms.
$$1x = 72$$
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Final Answer:
1) $a = \frac{47}{40}$ (or $1.175$)
2) $x = 23$
3) $x = -9.5$
4) $k = 6$
5) $x = -4$
6) $z = 107$
7) $x = -1.5$
8) $x = 72$
Parent Tip: Review the logic above to help your child master the concept of solving linear equations worksheet with answers.
