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Chapter 11: Problem 17

Solve. $$ 5 x+7 x=72 $$

Short Answer

Expert verified
x = 6.

Step by step solution

01

Combine like terms

First, add the terms with the variable on the left side of the equation. Combine 5x and 7x to get a single term with x.
02

Simplify the equation

The equation now becomes: \(12x = 72\).
03

Isolate the variable

Next, divide both sides of the equation by 12 to solve for x. \(x = \frac{72}{12}\).
04

Calculate the value of x

Perform the division to find the value of x. \(x = 6\).

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Key Concepts

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

Combine Like Terms
When solving linear equations, one of the initial and most essential steps is to **combine like terms**. This means we need to add or subtract terms that contain the same variable. In the exercise, we see the terms **5x** and **7x** on the left side of the equation. Both have the variable **x**, so they are like terms. By adding **5x** and **7x**, we combine them into a single term:

5\(x\) + 7\(x\) = 12\(x\).

This transformation compiles the equation into a more manageable and simplified form.
Simplify the Equation
After combining like terms, the next step is to **simplify the equation**. In this case, the equation becomes:

12\(x\) = 72

Simplifying the equation means we've reduced it to a format where it's easier to solve. Ensuring the equation is as straightforward as possible makes it easier to see what steps need to be taken next. We've eliminated the need to handle multiple terms and have created a single, clear equation.
Isolate the Variable
The following step in solving the linear equation is to **isolate the variable**. This means we need to get the term with the variable alone on one side of the equation. Here, our goal is to isolate **x**. We currently have 12\(x\) on the left side. To isolate **x**, we'll divide both sides of the equation by 12:

\[ x = \frac{72}{12} \]

This step ensures that the variable we are solving for (in this case, **x**) stands on its own, which sets us up to find its specific value.
Perform Division
Finally, to solve for **x**, we need to **perform the division** that we have established. It's straightforward: simply divide 72 by 12.
  • 72 divided by 12 equals 6
Therefore, our final value for **x** becomes:

\[ x = 6 \]

This final step is essential to arriving at the concrete solution for the variable. After performing the division, we have fully solved for **x** and completed the problem.

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