Problem 52 - Precalculus Student Solutions Manual 5th

Question

Solve each equation for \(x\). $$4 a-a x=3 b+b x$$

Step-by-Step Solution

Answer

x = \frac{4a - 3b}{b + a}

Step 1: Distribute coefficients

Distribute the coefficients on both sides of the equation. The equation is: \[4a - ax = 3b + bx\]No distribution needed here as the terms are already simplified.

Step 2: Move all terms involving x to one side

To isolate terms with x on one side, add ax on the left and subtract bx on the right:\[4a = 3b + bx + ax\]

Step 3: Factor x out of the equation

Factor x out from the terms on the right side:\[4a = 3b + x(b + a)\]

Step 4: Isolate x

Divide both sides of the equation by (b + a) to solve for x:\[x = \frac{4a - 3b}{b + a}\]

Key Concepts

Distributing Coefficients

In this step, we distribute coefficients to simplify the equation. Distributing means multiplying the coefficient outside the parentheses with each term inside. For example, in the equation \(4(a - x) = 4a - 4x\), the 4 is distributed to both 'a' and 'x'.
However, in our exercise, the terms are already simple: \(4a - ax = 3b + bx\). So, no distribution needed, and we can move to the next step.

Isolating Variables

Isolating a variable means getting all instances of it on one side of the equation. This helps us solve for that variable. In our problem, we first rearrange the terms:
\(4a - ax = 3b + bx\), we move all x terms to one side.
We do this through addition or subtraction. Here, we'll add \(ax\) to both sides:
\(4a = 3b + bx + ax\).
This puts all x terms on the right side, helping us get closer to isolating x.

Factoring

Factoring is used to simplify expressions by taking out the common factor. In the context of solving equations, this step helps isolate the variable we are solving for. For example, in the equation \(4a = 3b + bx + ax\), we see that both terms on the right side contain an 'x'.
We can factor out 'x' to make it easier to isolate. This gives us:
\(4a = 3b + x(b + a)\).
Now, we have x out of the parentheses, making it simpler to solve.

Balancing Equations

Balancing equations is a fundamental part of solving algebra problems. It's all about ensuring that what we do to one side of the equation, we do to the other. This keeps the equation 'balanced'.
For our equation: \(4a = 3b + x(b + a)\), to isolate x, we divide both sides by (b + a) to keep the equation balanced:
\(x = \frac{4a - 3b}{b + a}\).
This final step solves for x and completes the process of balancing the equation.