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Chapter 8: Problem 145

Solve the equations for the variable. $$12 x-17=-3 x+13$$

Short Answer

Expert verified
x = 2

Step by step solution

01

- Combine like terms

Add 3x to both sides of the equation to combine the terms with the variable x: \(12x - 17 + 3x = -3x + 13 + 3x\)This simplifies to \(15x - 17 = 13\)
02

- Isolate the variable term

Add 17 to both sides of the equation to isolate the term with the variable x on one side: \(15x - 17 + 17 = 13 + 17\)This simplifies to \(15x = 30\)
03

- Solve for the variable

Divide both sides of the equation by 15 to solve for x: \( \frac{15x}{15} = \frac{30}{15} \)This simplifies to \(x = 2\)

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Key Concepts

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

combining like terms
When solving linear equations, one of the first steps is to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the exercise given, we start with the equation:
\[12x - 17 = -3x + 13\]
We need to combine the terms with \(x\) to simplify the equation. We do this by adding \(3x\) to both sides of the equation:
\[12x - 17 + 3x = -3x + 13 + 3x\]
This gives:\[15x - 17 = 13\]
Now, the equation is simpler and only has one \(x\) term. This makes it easier to solve.
isolating the variable
The next step is to isolate the variable. This means you want to have the variable term alone on one side of the equation and the constant terms on the other side. In our example, after combining like terms, we have:
\[15x - 17 = 13\]
To isolate the variable term, add 17 to both sides of the equation to move the constant term to the right:
\[15x - 17 + 17 = 13 + 17\]
This simplifies to:\[15x = 30\]
Now, the variable term \(15x\) is isolated on the left side. This step is crucial because it sets up the equation for a straightforward solution.
division in equations
The final step is to solve for the variable. We do this by dividing both sides of the equation by the coefficient of the variable term. In our isolated equation:
\[15x = 30\]
The variable term has a coefficient of 15. To get the value of \(x\), divide both sides by 15:
\[ \frac{15x}{15} = \frac{30}{15} \]
This simplifies to:\[x = 2\]
By dividing, we have solved for \(x\). Division is often the last step in solving linear equations, making sure the variable stands alone.

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