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Chapter 8: Problem 294

In the following exercises, solve each equation. $$ 24 x+8 x-11 x=-7-14 $$

Short Answer

Expert verified
x = -1

Step by step solution

01

- Combine Like Terms

Combine the terms with the variable on the left-hand side of the equation: \(24x + 8x - 11x\).
02

- Simplify the Left Side

Simplify \(24x + 8x - 11x\) to get \(21x\). The equation is now: \(21x = -7 - 14\).
03

- Simplify the Right Side

Simplify \(-7 - 14\) to get \(-21\). The equation is now: \(21x = -21\).
04

- Solve for x

Divide both sides of the equation by 21 to isolate \(x\): \(x = \frac{-21}{21}\).
05

- Simplify

Simplify \(\frac{-21}{21}\) to get \(x = -1\).

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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 often to simplify the equation by combining like terms. Like terms are terms that contain the same variable raised to the same power. For example, in the equation provided, we have terms with the variable x: 24x, 8x, and -11x.

The goal is to combine these terms into one single term to make the equation simpler. When you add or subtract like terms, you only need to add or subtract their coefficients. The coefficients are the numerical parts that are multiplied by the variable. So, for our equation:
  • 24x + 8x - 11x
Combining these, you get:
  • (24 + 8 - 11)x = 21x
This combination of like terms sets up the equation to be easier to solve in the next steps.
Simplifying Expressions
After combining like terms, the next step in solving a linear equation is to simplify the expression further if needed. In our example:
  • 24x + 8x - 11x = -7 - 14
After combining the like terms, the equation becomes:
  • 21x = -7 - 14

Next, simplify the right side of the equation by performing the indicated operation:
  • -7 - 14 = -21
Now, the equation looks like this:
  • 21x = -21
Sometimes simplification involves removing parentheses, distributing, or combining constants. The key is to reduce the equation to a simpler form step-by-step, making it easier to isolate the variable.
Isolating Variables
The final goal in solving a linear equation is to isolate the variable on one side of the equation, which allows you to find its value. In our simplified equation:
  • 21x = -21
We need to isolate x. To do this, divide both sides by the coefficient of x, which is 21:
  • 21x / 21 = -21 / 21
This simplifies to:
  • x = -1

When isolating variables, remember:
  • Use operations that will cancel out the coefficients of the variable.
  • Always perform the same operation on both sides to maintain equality.
By isolating the variable, you effectively solve the equation and find the value of x.

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