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Chapter 2: Problem 9

Solve each linear equation. $$ 51+5(4-q)=56 $$

Short Answer

Expert verified
q = 3

Step by step solution

01

Distribute the 5

First, distribute the 5 in the equation to each term inside the parentheses: \( 5(4 - q) \).This gives us: \[ 51 + 20 - 5q = 56 \]
02

Combine like terms

Combine the constants on the left-hand side.This yields: \[ 71 - 5q = 56 \]
03

Isolate the variable term

Subtract 71 from both sides to isolate the term with the variable (\(-5q\)): \[ -5q = 56 - 71 \]Simplify the right-hand side: \(-5q = -15\)
04

Solve for q

Finally, divide both sides by \(-5\) to solve for \( q \): \[ q = \frac{-15}{-5} \]Simplify the fraction: \( q = 3 \)

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Key Concepts

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

Distributive Property
The Distributive Property is used to multiply a single term by each term inside a set of parentheses. For instance, in the equation given, you have:
$$ 51 + 5(4 - q) = 56 $$. Here, 5 is outside the parentheses. To apply the distributive property, you multiply 5 with each term inside the parentheses:
$$ 51 + 5 \times 4 - 5 \times q = 56$$.
When you carry out these multiplications, you get:
$$ 51 + 20 - 5q = 56 $$.
Combining Like Terms
Next, we combine like terms to simplify the equation. Like terms are terms that contain the same variables to the same power. In our example, combine all the constant numbers together:
  • Combine 51 and 20
This yields:
$$ 71 - 5q = 56 $$. By combining these like terms, we have a simplified version of the original equation.
Isolating Variables
Isolating the variable means getting the variable all by itself on one side of the equation. To isolate $$-5q$$, we subtract 71 from both sides:
$$ 71 - 5q - 71 = 56 - 71 $$ Simplifying both sides, we have:
$$ -5q = -15 $$.
This operation effectively isolates the term with the variable.
Solving for a Variable
Finally, to solve for the variable, you need to perform the necessary operations to obtain the variable by itself. This step involved dividing by the coefficient of $$q$$, which in this case is -5. Divide both sides by -5:
$$ q = \frac{-15}{-5} $$ Simplify the fraction:
$$ q = 3 $$.
You have now solved the equation, with $$ q $$ equaling 3.

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