Problem 44 - Algebra for College Students

Question

$$ \frac{x-a}{b}=\frac{y-a}{c} \text { for } y $$

Step-by-Step Solution

Answer

\( y = \frac{cx - ac + ab}{b} \)

Step 1: Cross Multiply

To eliminate the fractions, perform cross multiplication. Multiply the numerator of the first fraction by the denominator of the second and vice versa. This results in \( (x-a)c = (y-a)b \).

Step 2: Expand the Equation

Distribute the constants across the expressions in parentheses. Expand the equation to obtain \( cx - ac = by - ab \).

Step 3: Isolate the Term with y

We aim to solve for \( y \). Begin by moving terms not involving \( y \) to the other side of the equation. Add \( ab \) to both sides to get \( cx - ac + ab = by \).

Step 4: Solve for y

To isolate \( y \), divide every term by \( b \) on both sides of the equation. This gives \( y = \frac{cx - ac + ab}{b} \).

Key Concepts

Cross Multiplication

Cross multiplication is a powerful method used to eliminate fractions in equations with two fractions set equal to each other, like in the equation \(\frac{x-a}{b} = \frac{y-a}{c}\). It helps simplify the equation by removing the fractions, making further simplification easier.

To apply cross multiplication, you multiply the numerator of one fraction by the denominator of the other fraction. This technique can be visualized with a crisscross pattern:

• Multiply the numerator of the first fraction, \((x-a)\), by the denominator of the second fraction, \(c\).
• Then, multiply the numerator of the second fraction, \((y-a)\), by the denominator of the first fraction, \(b\).

This gives you the equation \((x-a)c = (y-a)b\). After cross multiplication, the fractions are gone, and the equation is easier to manage.

Isolation of Variables

Isolation of variables is an important technique for solving equations where you need to find the value of one particular variable, such as \(y\) in our example. The goal is to manipulate the equation until you have the variable you're solving for on one side by itself.

Once we have the equation \( cx - ac = by - ab \) after cross multiplying and expanding, we start isolating \(y\):

• First, get all the terms containing \( y \) on one side and the rest on the other side. You can do this by adding \( ab \) to both sides to ensure all terms without \( y \) are cleanly grouped together: \(cx - ac + ab = by \).

After that, you do the final step in isolating \( y \) by making sure it's alone.
Divide each term by \( b \), and you get \( y = \frac{cx - ac + ab}{b} \). Now, \(y\) is isolated, and you have solved the equation for \(y\).

Equation Expansion

Equation expansion involves multiplying out terms that are inside brackets to simplify or eliminate them, making the equation easier to solve. This can help reveal new terms that can be combined or isolated depending on what's needed.

In our solution, following the cross multiplication step, you get \( (x-a)c = (y-a)b \). The next step is to expand these terms by distributing the constants:

• Multiply \(c\) by both \(x\) and \(-a\) to obtain \(cx - ac\).
• Similarly, multiply \(b\) by both \(y\) and \(-a\) to get \(by - ab\).

The expanded equation becomes \(cx - ac = by - ab\). This is a more straightforward form that's ready for further simplification, such as isolating variables or combining like terms, allowing you to solve for the desired variables effectively.