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Chapter 4: Problem 180

Graph using the intercepts. \(-x+4 y=8\)

Short Answer

Expert verified
The x-intercept is (-8, 0) and the y-intercept is (0, 2). Plot these points and draw a line through them.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation \(- x + 4y = 8\) and solve for x. Substitute y = 0: \(- x + 4(0) = 8\). Simplifying, we get \( -x = 8 \). So, \( x = -8 \). The x-intercept is (-8, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation \(- x + 4y = 8\) and solve for y. Substitute x = 0: \( -0 + 4y = 8\). Simplifying, we get \( 4y = 8 \). So, \( y = 2 \). The y-intercept is (0, 2).
03

Plot the intercepts

Plot the x-intercept (-8, 0) and the y-intercept (0, 2) on a coordinate plane. These points are where the line crosses the x-axis and y-axis respectively.
04

Draw the line

Draw a straight line through the points (-8, 0) and (0, 2). This line represents the graph of the equation \(- x + 4y = 8\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

x-intercept
The x-intercept of a linear equation is the point where the line crosses the x-axis. To find the x-intercept, you set the y-value to 0 and solve for x. For example, in the equation \-x + 4y = 8\, if we set y to 0, the equation simplifies to \-x = 8\. Solving for x, we get x = -8. So, the x-intercept is the point \(-8, 0\). This means our line will touch the x-axis at x = -8, and this point is crucial for graphing the line.
y-intercept
The y-intercept is the point where the line crosses the y-axis. To find the y-intercept, you set the x-value to 0 and solve for y. Using the same equation \-x + 4y = 8\, if we set x to 0, the equation becomes \4y = 8\. Solving for y, we get y = 2. Therefore, the y-intercept is \(0, 2\). This indicates our line will intersect the y-axis at y = 2. Identifying the y-intercept helps anchor our graph on the coordinate plane.
coordinate plane
The coordinate plane is a two-dimensional surface defined by an x-axis (horizontal) and a y-axis (vertical). Each point on the plane is represented as \(x, y\) where x is the horizontal distance from the origin (0,0) and y is the vertical distance. For example, the point \(-8, 0\) lies 8 units to the left of the origin on the x-axis, and the point \(0, 2\) lies 2 units above the origin on the y-axis. Mastering the coordinate plane is essential for graphing any equation, as it allows you to plot points and visualize the relationships between them.
plotting points
Plotting points involves placing points on the coordinate plane based on their \(x, y\) coordinates. To plot a point, start at the origin and move along the x-axis to the x-coordinate value. From there, move vertically to the y-coordinate value. For example:
* **Point (-8, 0):** Start at the origin, move 8 units left on the x-axis and stay on the axis because y=0.
* **Point (0, 2):** Start at the origin, move 2 units up on the y-axis while staying at x=0.
Once you have these points plotted, you can draw a line through them to visualize the linear equation.

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