$ (9a + b) - (6a + b) = -9 - 5 \Rightarrow 3a = -14 \Rightarrow a = -\frac143 $ - Coaching Toolbox
Solve $ (9a + b) - (6a + b) = -9 - 5 $: A Step-by-Step Breakdown
Solve $ (9a + b) - (6a + b) = -9 - 5 $: A Step-by-Step Breakdown
In algebra, solving equations step-by-step not only reinforces mathematical concepts but also builds problem-solving confidence. Today, we’ll solve the equation:
$$
(9a + b) - (6a + b) = -9 - 5
$$
and derive that $ a = - rac{14}{3} $. This article walks you through each logical step to clearly understand how to isolate the variable $ a $ and find its exact value.
Understanding the Context
Step 1: Simplify Both Sides
Start by simplifying the expressions on both sides of the equation.
- The left-hand side:
$(9a + b) - (6a + b)$
Distribute the negative sign:
$9a + b - 6a - b$
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Key Insights
- The right-hand side:
$-9 - 5 = -14$
Now the equation becomes:
$$
9a + b - 6a - b = -14
$$
Step 2: Combine Like Terms
Combine like terms on the left side:
$ (9a - 6a) + (b - b) = 3a + 0 = 3a $
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So the equation simplifies to:
$$
3a = -14
$$
Step 3: Solve for $ a $
Divide both sides by 3:
$$
a = - rac{14}{3}
$$
Why $ b $ Cancel Out
Notice that $ b $ appears on both sides of the original equation: once inside the first parentheses as $ +b $, and again negatively as $ -b $. When simplified, $ +b - b = 0 $, effectively removing $ b $ from the equation. This highlights a crucial algebraic insight—terms that appear with opposite signs can cancel when simplifying.
Final Answer
$$
a = - rac{14}{3}
$$
