Problem 11 - 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. $$2 x-3 y=15$$

Step-by-Step Solution

Answer

The x-intercept of the graph is 7.5, and the y-intercept is -5.

Step 1: Finding the x-intercept

To find the x-intercept, set y equal to 0 in the equation. Then, solve the equation for x. That means for the equation \(2x - 3y = 15\), we will set y=0 and get \(2x - 3*0 = 15\). Solving for x gives \(x = 15/2 = 7.5\). So the x-intercept is 7.5.

Step 2: Finding the y-intercept

To find the y-intercept, set x equal to 0 in the equation. Then, solve the equation for y. So, in the equation \(2x - 3y = 15\), we will set x=0 to get \(-3y = 15\). Solving for y gives \(y = -15/3 = -5\). So the y-intercept is -5.

Key Concepts

X-Intercept Calculation

Let's dive into the world of linear equations by starting with the x-intercept calculation. This is an important anchor for graphing lines and understanding their behavior within a coordinate plane. The x-intercept is where a graph crosses the x-axis, which means the y-coordinate at this point is zero.
To find the x-intercept, we simply replace y with 0 in the equation and solve for x. For instance, let's take the equation \(2x - 3y = 15\). By putting y as 0, we get the equation \(2x = 15\). Dividing both sides by 2, we find \(x = 7.5\), which is our x-intercept. It is as straightforward as that!
Understanding this helps us visualize the point on the graph where the line will touch the x-axis, without needing to plot the entire line.

Y-Intercept Calculation

Similarly, the y-intercept is where the graph intersects with the y-axis, meaning the x-coordinate at this point is zero. It answers the question: where will the line cross the y-axis when we are not moving horizontally at all?
For the y-intercept calculation, you'll want to set x to zero in the original equation, then solve for y. Let's take our existing equation \(2x - 3y = 15\). Plugging in x as 0 gives us \(-3y = 15\). Divide both sides by -3, and voilà, we arrive at \(y = -5\). This is your y-intercept.
Knowing the y-intercept can be particularly useful when sketching graphs or interpreting the point where the line will start if drawn from the y-axis.

Linear Equations

Linear equations are the foundation of algebra and can be recognized by their straight line graphs. They are usually written in the form \(ax + by = c\), where a, b, and c are constants. The graph of such an equation will always be a straight line.
These lines have various properties, such as slope and intercepts, which help us understand their rate of change and where they position in the coordinate plane. For example, our equation \(2x - 3y = 15\) is linear because it adheres to this format, and plotting its x and y intercepts would help shape the line on a graph.
Grasping the concept of linear equations is essential, as it sets the stage for solving more complex problems in algebra and beyond.

Algebraic Solutions

Solving algebraic equations, such as our linear equation example, involves finding the values of the variables that make the equation true. These procedures can include a variety of operations, such as addition, subtraction, multiplication, division, and sometimes factoring.
Our main objective is to isolate the variable we're solving for—either x or y in the context of linear equations—and determine its value. The steps taken in our example—setting a variable to zero and then solving for the other—demonstrate systematic algebraic manipulation.
These algebraic solutions allow us to find intercepts without graphing, understand the structure of equations, and provide the insights necessary to analyze real-world situations mathematically.