Justify the steps in solving the equation by using the properties of equality.(2 points)

Question

2(4−8x)+7x=1−9x (Which Property)

8−16x+7x=1−9x (which property)


8−9x=1−9x (which property)

8−9x+9x=1−9x+9x (which property)


8=1

Question 2
(1 point)
How many solutions does this equation have?

Answer 1

To solve the equation \(2(4 - 8x) + 7x = 1 - 9x\), let's go through each step using the properties of equality and justify them:

1. Distributive Property:
   \(2(4 - 8x) + 7x = 1 - 9x\)
   Here, we use the Distributive Property to multiply 2 by both 4 and -8x. This leads to the next equation.

2. Combining Like Terms:
   \(8 - 16x + 7x = 1 - 9x\)
   In this step, we combined \(2 \cdot 4\) to get \(8\) and combined \(-16x\) and \(7x\) to get \(-16x + 7x = -9x\).

3. Combining Like Terms Again:
   \(8 - 9x = 1 - 9x\)
   Here, we simplified the left side by merging \(-16x + 7x\) into \(-9x\). This is also applying the property of combining like terms.

4. Adding Equal Terms:
   \(8 - 9x + 9x = 1 - 9x + 9x\)
   In this step, we added \(9x\) to both sides of the equation to eliminate the \(-9x\) term from the left side. This is an application of the Addition Property of Equality.

5. Simplifying:
   \(8 = 1\)
   After adding \(9x\) to both sides, we simplified to find that \(8\) is equal to \(1\). This is a contradiction.

Question 2:

How many solutions does this equation have?
The equation simplifies to \(8 = 1\), which is a false statement. This means there are no values of \(x\) that can satisfy the original equation. Therefore, the equation has no solutions.