Problem 12 - College Algebra

Question

For each of the following exercises, solve the equation for \(y\) in terms of \(x\) . $$3 x-2 y=6$$

Step-by-Step Solution

Answer

y = -\frac{3}{2}x + 3

Step 1: Isolate the term containing y

Start by subtracting the term involving x on the left-hand side from both sides of the equation. This gives us:\[ 3x - 2y = 6 \] Subtract \(3x\) from both sides:\[ -2y = 6 - 3x \]

Step 2: Solve for y

Divide both sides of the equation by -2 to isolate y:\[ y = \frac{6 - 3x}{-2} \]Simplify the equation:\[ y = -\frac{3}{2}x + 3 \]This gives y in terms of x.

Key Concepts

Solving for Variables

When tackling algebraic problems, solving for variables is like solving a mystery. You are given an equation with one or more unknowns, and your job is to find the value of these unknowns. In the given exercise, we are tasked with finding the value of \(y\) in terms of \(x\). To do this, we manipulate the equation to express \(y\) alone on one side.First, identify what you need to solve for. Here, it's \(y\). Equations often involve multiple variables, so make sure you clearly distinguish between them.

• Start by seeing what operations are applied to your variable, \(y\). For instance, is it being added, subtracted, multiplied, or divided by another number or variable?
• Undo these operations step by step, using inverse operations. This might mean subtracting something added, or dividing by something multiplied.
• Check your work throughout to avoid mistakes and ensure the resulting equation is correct.

Practicing a variety of problems will strengthen your skills in identifying the steps necessary to isolate and solve for a variable.

Linear Equations

Linear equations are like the building blocks of algebra. They represent relationships between variables in a straight line when graphed on a coordinate plane. Think of them as equations with no powers or square roots applied to the variables, just a simple relationship.A typical linear equation in two variables (\(x, y\)) can look like \(ax + by = c\). Here \(a\), \(b\), and \(c\) are constants and \(x\) and \(y\) are variables.

• Each solution to the equation corresponds to a point on the line defined by the equation when graphed.
• These equations cover simple, straightforward relationships. They can model real-life situations, like distance, speed, or cost.

In our exercise, the equation \(3x - 2y = 6\) is a linear equation, showing a simple relationship between \(x\) and \(y\). Understanding the properties of linear equations can help you solve them efficiently.

Isolation of Terms

Isolation of terms is a key step in solving equations. It involves rearranging the equation so that the variable of interest, \(y\) in our case, is alone on one side of the equation. This enables us to clearly see how it depends on other variables or constants in the equation.The basic process involves these steps:

• Move all terms not involving the variable you need to isolate to the other side of the equation. This often involves addition or subtraction.
• If your variable is being multiplied or divided by a coefficient, use division or multiplication to isolate the variable completely.
• Simplify the equation until the variable is by itself on one side.

In our example, we started with \(3x - 2y = 6\) and needed \(y\) by itself:1. Subtract \(3x\) from both sides: \[ -2y = 6 - 3x \]2. Divide each term by \(-2\) to solve for \(y\): \[ y = -\frac{3}{2}x + 3 \]This re-arrangement leaves us with \(y\) isolated. Let's you see the exact relationship between \(y\) and \(x\). Keep practicing examples to become fluent in isolating terms!