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Chapter 8: Problem 26

In the following exercises, solve each equation. $$ -6 x-11+7 x-5=-16 $$

Short Answer

Expert verified
x = 0

Step by step solution

01

- Combine Like Terms

Combine the like terms on the left side of the equation. Combine -6x + 7x = x so you get: x - 11 - 5 = -16
02

- Simplify the Equation

Simplify the constant terms on the left side of the equation: -11 - 5 = -16 so now you have: x - 16 = -16
03

- Isolate the Variable

Add 16 to both sides of the equation to isolate the variable x: x - 16 + 16 = -16 + 16 therefore you get: x = 0

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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 combine like terms. Like terms are terms that have the same variable raised to the same power. In the given equation ewline$$ -6x - 11 + 7x - 5 = -16 $$ ewlinelook for terms that have the same x variable.
  • -6x and 7x are like terms because they both have the variable x. Combine them by adding or subtracting their coefficients: ewline$$ -6x + 7x = x $$
After combining these terms, you simplify the equation to: ewline$$ x - 11 - 5 = -16 $$Combining like terms makes the equation easier to work with and is a critical step in simplifying linear equations.
Simplifying Equations
Once like terms are combined, the next step is to simplify the equation further by combining constant terms (numbers without variables). This simplifies the equation to its simplest form, making it easier to solve. Look at the simplified equation: ewline$$ x - 11 - 5 = -16 $$
  • -11 and -5 are constants. Combine them together: ewline$$ -11 - 5 = -16 $$
Now, the equation looks like this: ewline$$ x - 16 = -16 $$Combining constants helps simplify the equation and prepares it for the next step, which is isolating the variable.
Isolating Variables
The final step is to isolate the variable (x) on one side of the equation. This means you want x by itself. In the simplified equation: ewline$$ x - 16 = -16 $$ewlineadd 16 to both sides of the equation to isolate x: ewline$$ x - 16 + 16 = -16 + 16 $$ewlineSimplify both sides:
  • The left side simplifies to x because -16 and 16 cancel each other out: $$ x = 0 $$
Now you have x isolated on one side, and the solution to the equation is: ewline$$ x = 0 $$Isolating the variable helps you find the value of x, solving the equation completely.

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