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Chapter 2: Problem 192

Solve the following equations with variables on both sides. $$9 q=44-2 q$$

Short Answer

Expert verified
q = 4

Step by step solution

01

Move Variables to One Side

First, add 2q to both sides of the equation to move all variable terms to one side. This gives us: \[9q + 2q = 44\]
02

Combine Like Terms

Combine the terms with the variable q on the left side: \[11q = 44\]
03

Isolate the Variable

Divide both sides of the equation by 11 to isolate q: \[q = \frac{44}{11}\]
04

Simplify the Result

Simplify the right side of the equation: \[q = 4\]

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Key Concepts

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

variables on both sides
When solving linear equations, we often encounter variables on both sides of the equation. This means the unknown variable appears in terms on both the left and right sides. For example, in the equation $$ 9q = 44 - 2q $$, the term with the variable 'q' appears on both sides.
To solve this, we aim to bring all instances of the variable to one side.
This step makes the equation simpler to handle.
In our example, we add 2q to both sides to get the variables on one side: $$ 9q + 2q = 44 $$.
combining like terms
Once the variables are on the same side, look to combine like terms.
Like terms have the same variable raised to the same power.
For instance, in $$ 9q + 2q $$both terms have the variable 'q'.
Combining them involves adding their coefficients (the numeric part):
$$ 9q + 2q = 11q $$This process is crucial for simplifying the equation and moving towards isolating the variable.
isolating the variable
Isolating the variable means getting the variable alone on one side of the equation.
In our simplified equation $$ 11q = 44 $$,
we want 'q' by itself on one side.
This involves an inverse operation.
Since 'q' is multiplied by 11, we divide both sides by 11: $$ q = \frac{44}{11} $$
This step often involves either multiplication, division, addition, or subtraction, depending on the equation.
simplifying equations
Finally, simplify the equation to find the solution.
After isolating 'q', we have: $$ q = \frac{44}{11} $$
Simplification gives us the value: $$ q = 4 $$
This is the solution to our original equation!
Simplifying often includes reducing fractions or performing basic arithmetic operations to find the simplest form of the expression.

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