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

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 $

Continue Reading

Substitute \( a = 1 \) back into Equation 4:
3(1) + b = 5 \\
+ b = 5 \\

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Final Thoughts

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

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