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Chapter 2: Problem 36

Solve each linear equation. Show your work and check your answer. $$3(x-13)-x=9$$

Short Answer

Expert verified
The solution is \(x = 24\).

Step by step solution

01

Distribute

First, distribute the 3 across the terms inside the parenthesis: \(3(x - 13) - x = 9\)\(3x - 39 - x = 9\)
02

Combine Like Terms

Combine like terms on the left side of the equation: \(3x - x - 39 = 9\)\(2x - 39 = 9\)
03

Isolate the Variable Term

Add 39 to both sides of the equation to isolate the term with the variable: \(2x - 39 + 39 = 9 + 39\)\(2x = 48\)
04

Solve for the Variable

Divide both sides by 2 to solve for \(x\): \(\frac{2x}{2} = \frac{48}{2}\)\(x = 24\)
05

Check Your Answer

Substitute \(x = 24\) back into the original equation to verify: \(3(24 - 13) - 24 = 9\)\(3 \times 11 - 24 = 9\)\(33 - 24 = 9\)\(9 = 9\)Since both sides of the equation are equal, the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a way to simplify an expression by multiplying a single term across terms in parentheses. It's like sharing items equally between groups. Consider the equation: 3(x-13)-x=9. Using the distributive property, we multiply 3 by both x and -13. It looks like this: 3(x) - 3(13). This step transforms the equation into: 3x - 39 - x = 9.
Combining Like Terms
After distributing, we combine like terms. Like terms are expressions that contain the same variables raised to the same power. In our equation, 3x - x stands out as like terms. Combining them means we add or subtract their coefficients (the numbers in front of the variables). Here, 3x - x results in 2x. Now, our equation is simplified to: 2x - 39 = 9.
Isolating the Variable
The goal is to get the variable (x) alone on one side of the equation. This helps us find its value. Start by eliminating constants on the variable side. In 2x - 39 = 9, we add 39 to both sides to give 2x - 39 + 39 = 9 + 39. This simplifies to 2x = 48. Now, we divide both sides by 2 to isolate x, resulting in: x = 24.
Checking Solutions
Always verify your solution by substituting it back into the original equation. Replace x with 24 in 3(24-13)-24=9. Calculate each step: 3(11)-24 = 9, which simplifies to 33-24 = 9. Since both sides equal 9, our solution x=24 is correct. Remember, checking your answer ensures that the solution is accurate and reliable.

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