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Chapter 2: Problem 32

Solve each equation. $$6 x+8-5 x+16=32$$

Short Answer

Expert verified
x = 8

Step by step solution

01

Combine Like Terms

Combine the terms involving the variable on the left side of the equation: 6x - 5x + 8 + 16 = 32
02

Simplify the Equation

Simplify the left side by combining like terms: 6x - 5x becomes x and 8 + 16 becomes 24. So the equation now is: x + 24 = 32
03

Isolate the Variable

Subtract 24 from both sides to isolate x: x + 24 - 24 = 32 - 24 x = 8

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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, the first step often involves combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, in the equation
\(6x + 8 - 5x + 16 = 32\), the terms \(6x\) and \(-5x\) are like terms because they both involve the variable \(x\).
To combine like terms:
  • Add or subtract the numerical coefficients (the numbers in front of the variable). In this example, \(6x\) and \(-5x\) combine to become \(x\).
  • Combine constants (numbers without variables) separately. Here, \(8\) and \(16\) are combined to get \(24\).

So, after combining like terms, the equation simplifies to \(x + 24 = 32\).
Simplifying Equations
After combining like terms, the next step is to simplify the equation. Simplifying means reducing the equation to a simpler form. In our example, once the like terms are combined, we get:
\(x + 24 = 32\)
At this stage, simplifying the equation means ensuring that all like terms are collected together and the equation is straightforward to solve. Here, there are no further terms to combine, so the equation is already simplified. If there were more terms to combine or simplify, such steps would happen here.
Simplifying makes it easier to proceed to the next step of solving the equation.
Isolating the Variable
Finally, to find the value of the variable, you need to isolate it. Isolating the variable means getting the variable by itself on one side of the equation. In the simplified equation \(x + 24 = 32\), we want \(x\) alone on one side.
Here's how to isolate the variable:
  • Subtract \(24\) from both sides of the equation. This step ensures that \(x\) stands alone: \(x + 24 - 24 = 32 - 24\)
  • Perform the arithmetic to clean up the equation: \(x = 8\).

Now, \(x\) is isolated, and we find that \(x = 8\). This solution satisfies the original equation.

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Most popular questions from this chapter

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