Question

8k = -16
k = -2
10 given: -8(w + 1) = -5(w + 10); prove: w = 14
statements\treasons

given: 14 - 2(x + 8) = 5x - (3x - 34); prove: x = -9

Explanation:

For proving \( w = 14 \) from \( -8(w + 1) = -5(w + 10) \):

Step 1: Expand both sides

We use the distributive property \( a(b + c)=ab+ac \) to expand the left - hand side \( -8(w + 1) \) and the right - hand side \( -5(w + 10) \).
\( -8(w + 1)=-8w-8 \) and \( -5(w + 10)=-5w - 50 \), so the equation becomes \( -8w-8=-5w - 50 \).

Step 2: Add \( 8w \) to both sides

To get all the \( w \) terms on one side, we use the addition property of equality (adding the same value to both sides of an equation keeps it balanced).
\( -8w-8 + 8w=-5w - 50+8w \), which simplifies to \( -8 = 3w-50 \).

Step 3: Add 50 to both sides

Using the addition property of equality again, we add 50 to both sides to isolate the term with \( w \).
\( -8 + 50=3w-50 + 50 \), so \( 42 = 3w \).

Step 4: Divide both sides by 3

Using the division property of equality (dividing both sides of an equation by the same non - zero number keeps it balanced), we divide both sides by 3.
\( \frac{42}{3}=\frac{3w}{3} \), which gives \( w = 14 \).

Step 1: Expand both sides

First, expand \( -2(x + 8) \) using the distributive property \( a(b + c)=ab + ac \), so \( -2(x + 8)=-2x-16 \). Then, simplify the right - hand side: \( 5x-(3x - 34)=5x-3x + 34 = 2x+34 \) (using the rule \( a-(b - c)=a - b + c \)). The original equation becomes \( 14-2x-16=2x + 34 \).

Step 2: Combine like terms

Combine the constant terms on the left - hand side: \( 14-16=-2 \), so the equation is \( -2-2x=2x + 34 \).

Step 3: Add \( 2x \) to both sides

Using the addition property of equality, add \( 2x \) to both sides: \( -2-2x+2x=2x + 34+2x \), which simplifies to \( -2 = 4x+34 \).

Step 4: Subtract 34 from both sides

Using the subtraction property of equality (which is a form of the addition property, adding a negative number), subtract 34 from both sides: \( -2-34=4x+34-34 \), so \( -36 = 4x \).

Step 5: Divide both sides by 4

Using the division property of equality, divide both sides by 4: \( \frac{-36}{4}=\frac{4x}{4} \), which gives \( x=-9 \).

Answer:

By following the steps of expanding, using the addition property of equality, and using the division property of equality, we have proven that \( w = 14 \) when \( -8(w + 1)=-5(w + 10) \).

For proving \( x=-9 \) from \( 14-2(x + 8)=5x-(3x - 34) \):