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

Question

2(4−8x)+7x=1−9x

8−16x+7x=1−9x (which property)


8−9x=1−9x

8−9x+9x=1−9x+9x (which property)


8=1
Question 2
(1 point)
How many solutions does this equation have?

Answer 1

To justify the steps in solving the equation \(2(4-8x)+7x=1-9x\) using the properties of equality, we can examine each step you provided:

1. From: \(2(4−8x)+7x=1−9x\) To: \(8−16x+7x=1−9x\)
   Justification: This step uses the Distributive Property, which allows for distributing the \(2\) to both terms inside the parentheses, leading to \(2 \cdot 4 - 2 \cdot 8x\), resulting in \(8 - 16x + 7x = 1 - 9x\).

2. From: \(8−16x+7x=1−9x\)
   To: \(8−9x=1−9x\)
   Justification: This step combines like terms on the left side. By combining \(-16x\) and \(+7x\), we apply the Combining Like Terms property (a simplification step), resulting in \(8 - 9x = 1 - 9x\).

3. From: \(8−9x=1−9x\)
   To: \(8=1\)
   Justification: This step adds \(9x\) to both sides of the equation, showing the application of the Addition Property of Equality. This property states that if two expressions are equal, adding the same quantity to both sides will maintain the equality.

Now, regarding Question 2, to determine how many solutions this equation has:

From the final statement derived, \(8 = 1\) is a contradiction. Since this statement is not true (8 does not equal 1), it indicates that the original equation has no solutions.

Thus, the answer is that the equation has 0 solutions.