Question

Solve \(a x+b y=c\) for \(y\). A. \(y=\frac{c-a x}{b}\) B. \(y=\frac{b}{c-a x}\) C. \(y=\frac{c-b}{a x}\) D. \(y=\frac{c+a x}{b}\)

Short answer

A. \(y = \frac{c - a x}{b}\)

Step-by-step solution

Isolate the term with variable y

Start with the equation given: \(a x + b y = c\). To isolate the term with \(y\), subtract \(ax\) from both sides of the equation:\[a x + b y - a x = c - a x\]. This simplifies to \[b y = c - a x\].

Solve for y

To solve for \(y\), divide both sides of the isolated equation by \(b\): \[b y = c - a x\].Divide by \(b\) to isolate \(y\): \[ y = \frac{c - a x}{b}\].

Key concepts

Variable Isolation

When solving linear equations, one of the most crucial steps is isolating the variable of interest. Here, our goal is to isolate the variable y.
We start with the original equation: ax + by = c.To isolate y, we move all other terms to the opposite side of the equation. This involves algebraic manipulation, which means rearranging the terms while keeping the equation balanced.
In this case, we first subtract ax from both sides: ax + by - ax = c - ax.
This simplifies to by = c - ax.
Now, the term with y is isolated, and we can proceed to solve for y by further simplifying.

Linear Equations

Linear equations involve expressions of the first degree, meaning the variables are not squared or cubed, etc. They take the form ax + by = c, where a, b, and c are constants.
These equations depict a straight line when plotted on a graph and are foundational in algebra.
Understanding how to manipulate and solve linear equations is key for many mathematical concepts.
Our goal often includes solving for a specific variable, like y in this problem. To find y, appropriate algebraic steps need to be taken to rearrange the equation.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations using operations like addition, subtraction, multiplication, and division.
This skill is critical in isolating variables and finding their values.
Let's look at our equation after isolating the y term: by = c - ax.
The next step is to solve for y by dividing both sides by b: by/b = (c - ax)/b.
This simplifies to y = (c - ax)/b.
Through careful algebraic manipulation, we rearrange the given equation to solve for y.

GED Math

The GED (General Educational Development) test includes a mathematics section that covers various topics, including solving linear equations.
Being adept at variable isolation and algebraic manipulation is crucial for success.
This problem is a prime example of what you might encounter on the GED math test.
Understanding the steps to isolate and solve for a variable will help you tackle similar problems efficiently and accurately.
Consistent practice with these concepts will not only prepare you for the test but also build a solid foundation for more advanced mathematical topics.