Problem 18 - Introductory and Intermediate Algebra for College Students 4th

Question

Find the \(x\) -intercept and the \(y\) -intercept of the graph of each equation. Do not graph the equation. $$2 x=4 y-13$$

Step-by-Step Solution

Answer

So, the x-intercept is -6.5 and the y-intercept is 3.25

Step 1: Finding the x-intercept

Set y to 0 in the given equation: \(2x = 4(0) - 13\). Simplify and solve the equation to find the value of x.

Step 2: Finding the y-intercept

Set x to 0 in the given equation: \(2(0) = 4y -13\). Simplify and solve the equation to find the value of y.

Key Concepts

Algebra: Understanding Intercepts

In algebra, intercepts are key concepts when dealing with linear equations. They represent the points where the line crosses the axes on a graph. There are two main types of intercepts:

• x-intercept: The point where the line crosses the x-axis. This occurs when the value of y is zero.
• y-intercept: The point where the line crosses the y-axis. This happens when the value of x is zero.

To find these intercepts from an equation, you substitute the respective value (either x or y) with zero and solve for the other variable. This simple method transforms potentially complex algebraic expressions into straightforward values that highlight where lines meet axes on a graph.
By understanding intercepts, you're learning to decode the relationship between algebraic equations and their graphic representations. This decoding helps in visualizing how numbers behave under various mathematical situations. It's like finding the basic building blocks of the equation.

Linear Equations: Basic Form and Intercepts

Linear equations form the backbone of many algebraic concepts. A linear equation in its simplest form is expressed as \[ y = mx + b \], where \( m \) is the slope and \( b \) is the y-intercept.
This equation represents a straight line when graphed on a coordinate plane.In a typical linear equation like the one given: \[ 2x = 4y - 13 \], you can rearrange this to make it more recognizable. Rearranging helps identify the intercepts more easily.
By isolating either variable, you simply set the other to zero to uncover the intercepts.

• For the x-intercept, set \( y = 0 \) and solve for \( x \).
• For the y-intercept, set \( x = 0 \) and solve for \( y \).

This simple manipulation allows you to see how linear equations interact with graph axes.
Linear equations thus serve as a clear method to describe a direct relationship between two quantities.

Graphing Equations: Visualizing Linear Relationships

Graphing equations is an essential skill in understanding algebra's practical side. When you graph a linear equation, you visually represent the relationship between x and y. This direct visual allows you to grasp abstract concepts more concretely. Using the intercepts provides an easy way to plot these equations.
When you derive the intercepts from an equation, you determine two key points that the graph passes through:

• x-intercept: Located at (x,0)
• y-intercept: Located at (0,y)

These points are usually plotted first because they are easy to compute and provide a starting guide for the line's direction. Drawing a straight line through these intercepts gives you the graph of the equation.
This method is both simple and effective, making it a favorite technique for students new to graphing. By consistently using intercepts to plot graphs, you develop a clearer understanding of how equations translate into visual forms. It's a powerful skill illustrating the practical utility of algebra.