Problem 32 - Algebra for College Students

Question

For Problems \(31-44\), solve each equation for the indicated variable. $$ y=\frac{3}{4} x-\frac{2}{3} \text { for } x $$

Step-by-Step Solution

Answer

\(x = \frac{4}{3}y + \frac{8}{9}\)

Step 1: Isolate the Variable Term

In order to solve for \(x\), we need to isolate the term containing \(x\) on one side of the equation. The equation given is \(y = \frac{3}{4}x - \frac{2}{3}\). We will add \(\frac{2}{3}\) to both sides of the equation to get: \(y + \frac{2}{3} = \frac{3}{4}x\).

Step 2: Solve for the Variable

Now that we have \(y + \frac{2}{3} = \frac{3}{4}x\), we need to solve for \(x\) by getting rid of the \(\frac{3}{4}\) coefficient. We do this by multiplying both sides of the equation by \(\frac{4}{3}\): \(\frac{4}{3}(y + \frac{2}{3}) = x\).

Step 3: Simplification

Finally, simplify the expression \(\frac{4}{3}(y + \frac{2}{3})\) to express \(x\) clearly. This becomes: \(x = \frac{4}{3}y + \frac{4}{3}\times\frac{2}{3}\). Simplifying the second term, \(\frac{4}{3} \times \frac{2}{3} = \frac{8}{9}\). So, \(x = \frac{4}{3}y + \frac{8}{9}\).

Key Concepts

Isolation of Variable

The first step in solving a linear equation is isolating the variable. Here, our goal is to solve the equation \( y = \frac{3}{4}x - \frac{2}{3} \) for \( x \).
This means we need \( x \) on one side of the equation by itself. To start, let's look at the equation:

• The left side is \( y \).
• The right side is \( \frac{3}{4}x - \frac{2}{3} \).

To isolate \( x \), we first need to eliminate the constant term \( -\frac{2}{3} \) from the right side. We can do this by adding \( \frac{2}{3} \) to both sides of the equation:
\[ y + \frac{2}{3} = \frac{3}{4}x \]This step moves us closer to having \( x \) entirely by itself on one side of the equation.

Algebraic Manipulation

With the term \( \frac{3}{4}x \) now isolated on one side, the next step is to solve for \( x \) through algebraic manipulation.
Currently, our equation reads:
\[ y + \frac{2}{3} = \frac{3}{4}x \]

• To solve for \( x \), we need to get rid of the coefficient \( \frac{3}{4} \) in front of \( x \).
• We do this by multiplying both sides of the equation by the reciprocal of \( \frac{3}{4} \), which is \( \frac{4}{3} \).

This gives us:
\[ \frac{4}{3}(y + \frac{2}{3}) = x \]Multiplying by \( \frac{4}{3} \) effectively cancels out the \( \frac{3}{4} \) on the right side, leaving \( x \) by itself.

Equation Simplification

Once we've manipulated the equation to isolate \( x \), our final step is to simplify the expression. At this stage, we have:
\[ x = \frac{4}{3}(y + \frac{2}{3}) \] Simplification involves distributing \( \frac{4}{3} \) across the terms inside the parentheses:

• First, multiply \( \frac{4}{3} \) by \( y \) to get \( \frac{4}{3}y \).
• Next, multiply \( \frac{4}{3} \) by \( \frac{2}{3} \) to get \( \frac{8}{9} \).

Therefore, the simplified expression for \( x \) is:
\[ x = \frac{4}{3}y + \frac{8}{9} \] This clear expression shows \( x \) as a function of \( y \) and completes the process of solving the equation.