Problem 44 - Algebra for College Students

Question

Solve each equation for the indicated variable. \(\frac{x-a}{b}=\frac{y-a}{c}\) for \(y\)

Step-by-Step Solution

Answer

The solution for \(y\) is \(y = \frac{cx + ba - ca}{b}\).

Step 1: Clear the Fractions

To eliminate the fractions from the equation, cross-multiply. This gives us \[ c(x - a) = b(y - a) \]

Step 2: Distribute and Expand

Distribute the coefficients \(c\) and \(b\) across the terms in the parentheses:\[ cx - ca = by - ba \]

Step 3: Isolate Terms with y

To isolate the term containing \(y\), move \(-ba\) to the left side of the equation by adding \(ba\) to both sides:\[ cx - ca + ba = by \]

Step 4: Solve for y

Divide both sides of the equation by \(b\) to solve for \(y\):\[ y = \frac{cx + ba - ca}{b} \]

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to simplify equations involving fractions, particularly when there are two ratios equated against each other. In our exercise, we have the equation \(\frac{x-a}{b}=\frac{y-a}{c}\).

• Think of each side of the equation as a fraction. The goal here is to eliminate these fractions by multiplying across the equals sign.
• Multiply the numerator of one fraction by the denominator of the other, and vice versa, to form a new equation without fractions.

For instance, when we apply cross-multiplication to our equation, we get \(c(x-a) = b(y-a)\). This step is crucial as it simplifies the equation, allowing us to work with more straightforward algebraic terms. Cross-multiplication makes it easier to isolate variables later on.

Clearing Fractions

Clearing fractions from an equation means manipulating the equation in such a way that no fractions remain. It allows for easier manipulation of the algebraic equation.

• Cross-multiplication, as described earlier, is one of the primary methods used to clear fractions.
• In our example, we eliminated the fractions in \(\frac{x-a}{b}=\frac{y-a}{c}\) by using cross-multiplication to generate \(c(x-a) = b(y-a)\).

Once the equation is free of fractions, it is easier to manage and brings us one step closer to solving for the unknown variable.

Isolating Variables

Isolating a variable means rearranging an equation in order to represent one specific variable on one side of the equation, typically leaving that variable by itself.

• After clearing the fractions from our equation, we have \(cx - ca = by - ba\).
• Our goal now is to isolate \(y\). This involves moving all terms not containing \(y\) to the other side of the equation: \(cx - ca + ba = by\).
• Once \(y\) is isolated, you can solve for it by performing additional operations such as dividing or factoring.

This step is critical in solving equations, as it simplifies the equation to find the value of the unknown variable, which in this case is \(y\).

Linear Equations

Linear equations are equations of the first degree, meaning they involve only terms that are constant or linear in nature; these terms do not have any exponents beyond one.

• In our exercise, the equation \(\frac{x-a}{b}=\frac{y-a}{c}\) is a linear equation once we apply cross-multiplication and simplify it to \(cx - ca = by - ba\).
• Linear equations are fundamental because they form a straight line when graphed, representing the simplest form of relationships in algebra.
• Solving a linear equation often involves isolating the variable of interest, a process we have applied to solve for \(y\) by following operations like addition, subtraction, and division.

Understanding linear equations is key to delving deeper into algebra, as they serve as a foundation for understanding more complex mathematical relationships.