Problem 23 - Precalculus with Limits

Question

In Exercises 23-32, find the \( x \)- and \( y \)-intercepts of the graph of the equation. \( y = 5x - 6 \)

Step-by-Step Solution

Answer

The x-intercept and y-intercept of the equation \( y = 5x - 6 \) are \( x = 6/5 \) or \( 1.2 \) in decimal form and \( y = -6 \) respectively.

Step 1: Find the x-intercept

Set \( y = 0 \) in the equation and solve for \( x \). Hence, we have:0 = 5x - 6 This implies that \( 5x = 6 \)Dividing both sides by 5 gives us \( x = 6/5 \). Thus the x-intercept is \( 6/5 \) or \( 1.2 \) when represented in decimal form.

Step 2: Find the y-intercept

Set \( x = 0 \) in the equation and solve for \( y \). Hence, we have:\( y = 5*0 - 6 \)This means that \( y = -6 \). Thus the y-intercept is -6.

Key Concepts

X-Intercept

The x-intercept of a graph is the point where the line crosses the x-axis. To find the x-intercept of a linear equation like the one in the given exercise, you set the y-value to zero and solve for x. This is because any point on the x-axis has a y-value of zero. In mathematical terms, you are looking for the value of x when y equals zero.

For the equation \( y = 5x - 6 \), setting y to zero gives us \( 0 = 5x - 6 \). Solving for x, we find that x equals \( \frac{6}{5} \) or in decimal form, 1.2. Therefore, the x-intercept, where the line cuts through the x-axis, is at the point (1.2, 0).

Y-Intercept

Conversely, the y-intercept is where the line crosses the y-axis. To find it, you set the x-value to zero and then solve for y. Every point on the y-axis has an x-value of zero. For the same equation \( y = 5x - 6 \), by setting x to zero, we get \( y = 5*0 - 6 \), which simplifies to \( y = -6 \). Hence, the y-intercept is at the point (0, -6), signifying where the line meets the y-axis.

Finding both intercepts is crucial for graphing linear equations, as they provide two points through which the line passes, creating a visual representation of the equation on the coordinate plane.

Solving Linear Equations

The process of solving linear equations typically involves isolating the variable you want to solve for. When dealing with linear equations in two variables like \( y = mx + b \), where m is the slope and b is the y-intercept, you often have to make one variable subject to get specific points. In the context of our example, finding the x- and y-intercepts involves setting one variable to zero and solving for the other.

This foundational concept not only helps in graphing but also in understanding how variables relate to each other and to the overall structure of the equation. For students, mastering these skills is essential for algebra, calculus, and beyond.

Coordinate Geometry

The field of coordinate geometry, also known as analytic geometry, deals with the graphing of equations on the coordinate plane and the study of geometric figures through algebraic representation. In this context, understanding and finding x- and y-intercepts form a core part of this subject area.

The ability to graph linear equations like \( y = mx + b \) requires knowing how to find these intercepts, as they mark key points where the line crosses the axes. These intercepts can also tell you a great deal about the equation, such as the slope and where the graph is positioned in the plane. Developing a good grasp of coordinate geometry is essential for solving more complex problems, including those involving curves, circles, and other geometric figures.