Problem 63 - Algebra A Combined Function

Question

Solve each equation for y. See Section 2.5. $$ 2 x+4 y=5 $$

Step-by-Step Solution

Answer

y = \frac{5 - 2x}{4}

Step 1: Identify the Equation

The given equation is \(2x + 4y = 5\). Our goal is to solve for \(y\).

Step 2: Isolate the Term with y

Subtract \(2x\) from both sides of the equation to isolate the terms with \(y\) on one side. This results in \(4y = 5 - 2x\).

Step 3: Solve for y

Divide every term by 4 to isolate \(y\) completely: \( y = \frac{5 - 2x}{4} \).

Key Concepts

Solving for a Variable

When tackling algebraic equations, one common task is "solving for a variable." This means isolating a chosen variable to one side of the equation, representing its relationship with other terms. In the context of our exercise, we're aiming to solve for the variable \( y \) in the equation \( 2x + 4y = 5 \).

The primary objective is to express \( y \) in terms of \( x \), which provides a way to see how changes in \( x \) affect \( y \). It turns equations into useful tools for predicting outcomes. Understanding this process will help in handling more complex algebraic problems involving multiple variables.

Linear Equations

A linear equation is an equation that forms a straight line when graphed on a coordinate plane. The general form of a linear equation in two variables is \( Ax + By = C \), where \( A \), \( B \), and \( C \) are constants. Our original exercise, \( 2x + 4y = 5 \), fits this description.

Linear equations are fundamental in mathematics because they show constant rates of change. In everyday problems, linear equations allow us to model real-world situations like speed, cost, or distance. They are easy to manipulate and understand, so mastering them provides a strong foundation for more advanced topics.

By learning to solve linear equations, like in our task, you gain insights into their geometric representations – understanding how they graph and their intersections with other lines.

Isolating Terms

Isolating terms is a critical step when solving equations. It involves rearranging the equation to group terms, often onto separate sides of the equation. The goal is to simplify until the desired variable stands alone.

In the exercise \( 2x + 4y = 5 \), the task is to isolate \( y \). We start by subtracting \( 2x \) from both sides, focusing on moving anything not involving \( y \) to the other side: now the equation reads \( 4y = 5 - 2x \). This split leaves the term with \( y \) on one side.

The final step to isolate \( y \) completely is dividing everything by 4, simplifying the equation to \( y = \frac{5 - 2x}{4} \). The process of isolating terms is crucial as it directly impacts the simplicity and readability of the final result, making further calculations or interpretations easier.