Question Solve the linear equation $9=5(x-2)-(x-7)$ and select the correct solution set.

Studdy Solution

STEP 1

Assumptions
1. We are given a linear equation $9 = 5(x - 2) - (x - 7)$ .
2. We need to solve for the variable $x$ .
3. The solution set will be determined based on the nature of the solution.

STEP 2

Expand the terms within the parentheses by applying the distributive property.
$9 = 5 \cdot (x - 2) - 1 \cdot (x - 7)$

STEP 3

Distribute the multiplication over the subtraction for both terms.
$9 = 5x - 10 - x + 7$

STEP 4

Combine like terms on the right side of the equation.
$9 = (5x - x) + (-10 + 7)$

STEP 5

Simplify the combined terms.
$9 = 4x - 3$

STEP 6

Add $3$ to both sides of the equation to isolate the term with the variable $x$ on one side.
$9 + 3 = 4x - 3 + 3$

STEP 7

Simplify both sides of the equation.
$12 = 4x$

STEP 8

Divide both sides of the equation by $4$ to solve for $x$ .
$\frac{12}{4} = \frac{4x}{4}$

STEP 9

Calculate the value of $x$ .
$x = 3$

STEP 10

The solution to the equation is $x = 3$ . We can check this solution by substituting $3$ back into the original equation.
$9 = 5(3 - 2) - (3 - 7)$

STEP 11

Simplify the terms within the parentheses.
$9 = 5(1) - (-4)$

STEP 12

Multiply and add the numbers.
$9 = 5 + 4$

STEP 13

Combine the numbers to see if they equal $9$ .
$9 = 9$

STEP 14

Since the left side of the original equation equals the right side when $x = 3$ , the solution checks out.
The solution set is $\{x \mid x = 3\}$ , which is not one of the provided choices. Therefore, we need to write the correct solution set in the answer box.
The correct choice is: The solution set is $\{3\}$ .

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