Problem 9 - College Algebra

Question

For each of the following exercises, find the \(x\)-intercept and the \(y\)-intercept without graphing. Write the coordinates of each intercept. $$3 x+8 y=9$$

Step-by-Step Solution

Answer

x-intercept: (3, 0); y-intercept: (0, 9/8).

Step 1: Introduction

To find the intercepts of the equation, we need to find the points where the line crosses the x-axis and the y-axis. The x-intercept is found by setting \( y = 0 \) and solving for \( x \). The y-intercept is found by setting \( x = 0 \) and solving for \( y \).

Step 2: Finding the x-intercept

For the x-intercept, set \( y = 0 \) in the equation \( 3x + 8y = 9 \). This gives us \( 3x + 8(0) = 9 \), which simplifies to \( 3x = 9 \). Solving for \( x \), we divide both sides by 3, yielding \( x = 3 \). Therefore, the x-intercept is \((3, 0)\).

Step 3: Finding the y-intercept

For the y-intercept, set \( x = 0 \) in the equation \( 3x + 8y = 9 \). This gives us \( 3(0) + 8y = 9 \), which simplifies to \( 8y = 9 \). Solving for \( y \), we divide both sides by 8, yielding \( y = \frac{9}{8} \). Therefore, the y-intercept is \( (0, \frac{9}{8}) \).

Key Concepts

Finding x-intercept

Intercepts in a linear equation like \(3x + 8y = 9\) are essential to understand as they tell us where the line will touch the axes. To find the x-intercept, we focus on the point where the line crosses the x-axis. At this crossing point, the y-coordinate is always zero. Therefore, we set \(y = 0\) in our equation. By substituting \(y = 0\) into \(3x + 8y = 9\), the equation simplifies to \(3x + 8(0) = 9\) or simply \(3x = 9\). Dividing both sides by 3 isolates \(x\), yielding \(x = 3\).
Now, we know the x-intercept is the point \((3, 0)\). This means when you visually plot this line, it will intersect the x-axis at this coordinate. Understanding this process helps you determine where the line interacts with the x-axis, which is crucial for graphing and analysis.

Finding y-intercept

The y-intercept of a line is the point where it crosses the y-axis. Unlike the x-intercept, at this crossing point, the x-coordinate is zero. To find the y-intercept, we set \(x = 0\) in our equation. This quick substitution shows where the line hits the y-axis.For our equation \(3x + 8y = 9\), setting \(x = 0\) leaves us with \(3(0) + 8y = 9\), simplifying to \(8y = 9\). By dividing both sides by 8, we solve for \(y\) and find \(y = \frac{9}{8}\).
This gives us the y-intercept as \((0, \frac{9}{8})\). This shows that the line touches the y-axis at \(y = \frac{9}{8}\). By understanding how the y-intercept is found, you gain a clearer grasp of how linear equations graphically interact with the y-axis.

Linear equation solutions

Solving linear equations involves finding values for variables that make the equation true. In the context of finding intercepts, we solve for one variable at a time while substituting zero for the other. This is a straightforward way to pinpoint where the line represented by the equation intersects the axes.The steps for solving an equation like \(3x + 8y = 9\) include:

• Finding the x-intercept by setting \(y = 0\). Substitute to simplify and solve for \(x\).
• Finding the y-intercept by setting \(x = 0\). Substitute to simplify and solve for \(y\).

Linear equations graph as straight lines, and finding these intercepts simplifies creating and understanding these graphs. Recognizing how these points are determined reinforces the connection between algebraic solutions and their geometric representations.