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Solve the equation:

[tex]\(7x - \frac{1}{2}(8x + 2) = 6\)[/tex]

[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Step} & \text{Work} & \text{Justification} \\
\hline
1 & 7x - 4x - 1 = 6 & \text{Distributive property} \\
\hline
2 & 3x - 1 = 6 & \text{Combine like terms} \\
\hline
3 & 3x = 7 & \text{Addition property of equality} \\
\hline
4 & x = \frac{7}{3} & \text{Division property of equality} \\
\hline
\end{array}
\][/tex]

Which of the following contains all of the correct justifications Kelsey used to solve this equation?

A. Distributive property, Combine like terms, Addition property of equality, Division property of equality

B. Multiplication property of equality, Combine like terms, Addition property of equality, Division property of equality

C. Distributive property, Combine like terms, Subtraction property of equality, Division property of equality

D. Multiplication property of equality, Combine like terms, Subtraction property of equality, Division property of equality



Answer :

GinnyAnswer
Let's work through the solution step-by-step:
[tex]$ 7x - \frac{1}{2}(8x + 2) = 6 $[/tex]

### Step 1: Apply the Distributive Property
We need to distribute [tex]\(\frac{1}{2}\)[/tex] across the expression inside the parentheses.
[tex]$ 7x - \frac{1}{2} \cdot 8x - \frac{1}{2} \cdot 2 = 6 $[/tex]
Simplifying this, we get:
[tex]$ 7x - 4x - 1 = 6 $[/tex]
Justification: Distributive property

### Step 2: Combine Like Terms
We combine the like terms [tex]\(7x\)[/tex] and [tex]\(-4x\)[/tex].
[tex]$ (7x - 4x) - 1 = 6 $[/tex]
This simplifies to:
[tex]$ 3x - 1 = 6 $[/tex]
Justification: Combine like terms

### Step 3: Move the Constant to the Other Side of the Equation
We add 1 to both sides of the equation to isolate the term with [tex]\(x\)[/tex] on one side:
[tex]$ 3x - 1 + 1 = 6 + 1 $[/tex]
This simplifies to:
[tex]$ 3x = 7 $[/tex]
Justification: Addition property of equality

### Step 4: Divide Both Sides by 3 to Isolate [tex]\(x\)[/tex]
Now, we divide both sides of the equation by 3 to solve for [tex]\(x\)[/tex]:
[tex]$ x = \frac{7}{3} $[/tex]
Justification: Division property of equality

Based on the detailed solution steps and their justifications, the correct list of justifications Kelsey used to solve the equation is:

1. Distributive property
2. Combine like terms
3. Addition property of equality
4. Division property of equality

Thus, the correct option is:
1. Distributive property 2. Combine like terms 3. Addition property of equality 4. Division property of equality

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