Let's complete the two-column algebraic proof to show that x = 7.
Reasons:
1. Given
2. Distributive property
3. Addition property of equality
4. Simplification
5. Subtraction property of equality
6. Division property of equality
Statements:
1. -8(x - 3) = -32
- This is the given equation, stating that eight times the quantity x minus 3 equals -32.
2. -8x + 24 = -32
- By applying the distributive property (multiply -8 with both x and -3), we get -8x + 24.
3. -8x + 24 + 32 = -32 + 32
- We add 32 to both sides of the equation to isolate the term containing x. This follows the addition property of equality.
4. -8x + 56 = 0
- By simplifying both sides, we add 24 and 32 together to get 56 on the left side, and since -32 + 32 equals 0, we get 0 on the right.
5. -8x = -56
- Subtract 56 from both sides of the equation to move all constants to the right side. This uses the subtraction property of equality.
6. x = 7
- Dividing both sides by -8 isolates x on the left side, and -56 divided by -8 equals 7. This step uses the division property of equality, which says that both sides can be divided by the same nonzero number without changing the equality.
So our detailed steps prove that x = 7 using algebraic properties.
