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

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

1 answer

Let's break down the steps of solving the equation \( 2(4 - 8x) + 7x = 1 - 9x \) and justify each step using the properties of equality:

  1. Distribute: The first step is to distribute the 2 on the left side: \[ 2(4 - 8x) + 7x \Rightarrow 8 - 16x + 7x \] Justification: This step uses the Distributive Property, which states that \( a(b + c) = ab + ac \).

  2. Combine like terms: The next step combines the like terms on the left side: \[ 8 - 16x + 7x = 8 - 9x \] Justification: This step applies the Commutative Property of Addition and Associative Property of Addition to rearrange and combine the coefficients of \( x \).

  3. Add or subtract terms: Now, we isolate the variable by moving \( -9x \) to the right side: \[ 8 - 9x = 1 - 9x \Rightarrow 8 - 9x + 9x = 1 - 9x + 9x \] Justification: This step uses the Addition Property of Equality, which states that if two expressions are equal, adding (or subtracting) the same value from both sides maintains the equality.

  4. Simplify: After simplifying, we find: \[ 8 = 1 \] Justification: This is the result of cancelling out the terms, which demonstrates that the equations do not equate.

Question 2

How many solutions does this equation have?

The equation resolves to \( 8 = 1 \), which is a contradiction (it is never true). Therefore, this equation has no solutions.