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Chapter 8: Problem 143

Solve the equations for the variable. $$9 c+7=-2 c-37$$

Short Answer

Expert verified
c = -4

Step by step solution

01

Combine Like Terms

First, we need to combine the like terms involving the variable. To do this, add \(2c\) to each side of the equation to get: \[9c + 2c + 7 = -37\].
02

Simplify the Equation

Combine the like terms on the left side. This gives: \[11c + 7 = -37\].
03

Isolate the Variable Term

Next, subtract 7 from both sides to isolate the term with the variable: \[11c = -37 - 7\].
04

Simplify Again

Simplify the right side of the equation to get: \[11c = -44\].
05

Solve for the Variable

Divide both sides by 11 to solve for \(c\): \[c = \frac{-44}{11}\].
06

Final Simplification

Simplify the fraction to get the final answer: \[c = -4\].

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Key Concepts

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

Combining Like Terms
Combining like terms is one of the essential steps in solving linear equations. Like terms are terms that contain the same variable raised to the same power. In our problem, we have terms involving the variable 'c' on both sides of the equation: \(9c + 7 = -2c - 37\). To combine these like terms, we need to get all the 'c' terms on one side. By adding \2c\ to each side, we get: \[9c + 2c + 7 = -37\]. This makes it easier to solve because now all the 'c' terms are combined into \[11c\], leaving the other side of the equation with just the constants.
Isolating Variables
Isolating the variable is a crucial part of solving linear equations. We need to get the variable term by itself on one side of the equation. In our exercise, after combining like terms, we have \[11c + 7 = -37\]. To isolate \-11c\, we subtract 7 from both sides: \[11c = -37 - 7\]. This simplifies to \11c = -44\. Now, the term with the variable 鈥榗鈥� is isolated, making it easier to solve for the variable.
Simplifying Equations
Simplifying equations is the process of making the equation as straightforward as possible. First, combining like terms and isolating variables help reduce complexity. From our exercise, we had \11c + 7 = -37\, which simplifies to \11c = -44\. Next, we solve for the variable by dividing by the coefficient in front of \'c\: \[c = \frac{-44}{11}\]. Simplifying that fraction gives us \[c = -4\]. Each simplification step breaks the problem down, making it easy to find the solution.

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