$ (9a + b) - (6a + b) = -9 - 5 \Rightarrow 3a = -14 \Rightarrow a = -\frac{14}{3} $

February 22, 2026

$(9a + b) - (6a + b) = -9 - 5 \Rightarrow 3a = -14 \Rightarrow a = -\frac{14}{3}$

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.

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: $9a + b - 6a - b$
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 $

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

$$ a = -rac{14}{3} $$

Additional Insight: Core Equation Meaning

The original equation $ (9a + b) - (6a + b) = -14 $ models a real-world comparison: the difference between two linear expressions in $ a $ and $ b $. This simplification eliminates variables where possible, demonstrating how strategic expansion and combination reduce complexity.

Summary

Distribute and simplify both sides.
Combine like terms to eliminate $ b $.
Solve directly for $ a $.
Double-check by substituting back into the original equation.

By mastering this step-by-step method, you gain clarity and precision in solving linear equations — essential skills for advanced math and STEM studies.

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