Justify the steps in solving the equation by using the properties of equality.

Question

2(4−8x)+7x=1−9x

8−16x+7x=1−9x


8−9x=1−9x

8−9x+9x=1−9x+9x


8=1

Question 2
How many solutions does this equation have?

Answer 1

Let's go through the steps of solving the equation \(2(4 - 8x) + 7x = 1 - 9x\) and justify each step using the properties of equality, as requested.

1. Distribute the \(2\) to the terms inside the parentheses:
   \[ 2(4 - 8x) + 7x = 1 - 9x \] This gives: \[ 8 - 16x + 7x = 1 - 9x \] Justification: This step uses the Distributive Property.

2. Combine like terms on the left side:
   \[ 8 - 16x + 7x = 1 - 9x \] This simplifies to: \[ 8 - 9x = 1 - 9x \] Justification: This step uses the property of combining like terms.

3. Add \(9x\) to both sides to isolate the constant:
   \[ 8 - 9x + 9x = 1 - 9x + 9x \] This simplifies to: \[ 8 = 1 \] Justification: This step is justified by the Addition Property of Equality, which states that adding the same value to both sides of an equation preserves equality.

Now, at this point in the solution, we find that: \[ 8 = 1 \] This is a contradiction, as \(8\) is not equal to \(1\).

Conclusion for Question 2

Since we reached a contradiction, it indicates that there are no solutions to the equation \(2(4 - 8x) + 7x = 1 - 9x\). Therefore, the equation has zero solutions.