Problem 15 - 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. $$7 x-9 y=0$$

Step-by-Step Solution

Answer

The x-intercept and the y-intercept of the given equation are both 0.

Step 1: Finding the x-intercept

The x-intercept is found by setting \(y = 0\) in the equation and solving for \(x\). So, substituting \(y = 0\) into the equation \(7x - 9y = 0\) gives \(7x = 0\). Solving for \(x\) gives \(x = 0\). So, the x-intercept is 0.

Step 2: Finding the y-intercept

The y-intercept is found by setting \(x = 0\) in the equation and solving for \(y\). So, substituting \(x = 0\) into the equation \(7x - 9y = 0\) gives \(-9y = 0\). Solving for \(y\) gives \(y = 0\). So, the y-intercept is 0.

Key Concepts

x-intercept

The concept of an x-intercept is straightforward once you familiarize yourself with it. An x-intercept is the point where the graph of an equation crosses the x-axis. This point has a y-coordinate of zero, since it's located directly on the x-axis. To find it, you substitute 0 for y in the linear equation. This effectively simplifies the equation and focuses the calculation on finding x.
In the example: \(7x - 9y = 0\), to find the x-intercept:

• Set \(y = 0\).
• The equation becomes \(7x = 0\).
• Solving for x gives \(x = 0\).

Thus, the x-intercept here is (0, 0). Remember, even if it seems like a math trick, it's always about getting that y to zero, so you're left with just x to solve for. This is essential in identifying where the graph crosses the x-axis.

y-intercept

Understanding the y-intercept is equally as important. A y-intercept is the point where the graph crosses the y-axis. This means the x-coordinate is zero at this point, as it lies directly on the y-axis. To find the y-intercept, you substitute 0 for x in the equation. This will simplify your calculations to focus solely on y.
For the equation \(7x - 9y = 0\):

• Set \(x = 0\).
• Substituting gives \(-9y = 0\).
• Solving for y leads to \(y = 0\).

Hence, the y-intercept is also (0, 0) in this instance. Recognizing this, you understand that the intercept is simply where the graph touches the vertical axis, critical for understanding the graph's behavior.

linear equation

Linear equations are fundamental in algebra and graphing. They represent a straight line when graphed on a coordinate plane. The general form of a linear equation is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. These equations can be manipulated to find various properties of the straight line they represent, such as slopes and intercepts.
In our case, we have: \(7x - 9y = 0\). This is a particular case where whenever the constants of the equation permit, both intercepts were found at the origin, indicating that the line passes through (0, 0). Each term in a linear equation represents a direction or inclination of the line. Finding intercepts is a crucial step for a clearer understanding of its characteristics.

• The equation defines a relationship between x and y.
• Every solution (x, y) plotted is a point on the line.
• Intercepts tell us where the line crosses the respective axes.

This simplicity and directness of linear equations make them excellent for modeling real-world situations. By focusing on intercepts, you unlock a deeper insight into this equation type.