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Chapter 11: Problem 156

In the following exercises, graph using the intercepts. $$ -x+4 y=8 $$

Short Answer

Expert verified
x-intercept: -8, y-intercept: 2. Draw a line through these points.

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \( y = 0 \) and solve for \( x. \) Starting with the equation \(-x + 4y = 8,\) substitute \( y = 0. \): \(-x + 4(0) = 8 \Rightarrow -x = 8 \Rightarrow x = -8. \) Thus, the x-intercept is \(-8. \)
02

Find the y-intercept

To find the y-intercept, set \( x = 0 \) and solve for \ y. \ Starting with the equation \(-x + 4y = 8,\) substitute \( x = 0. \): \( -0 + 4y = 8 \Rightarrow 4y = 8 \Rightarrow y = 2. \) Thus, the y-intercept is \ 2. \
03

Plot the intercepts on the graph

Plot the x-intercept \((-8, 0)\) and the y-intercept \((0, 2)\) on a coordinate plane.
04

Draw the line

Draw a straight line through the points \((-8, 0)\) and \((0, 2).\) This is 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 where the line crosses the x-axis on a coordinate plane. To find this intercept, set the value of y to zero in the given equation, and then solve for x. In our exercise, we started with the equation \(-x + 4y = 8\). By substituting y = 0, the equation simplifies to: \-x + 4(0) = 8, which leads to \-x = 8. Solving for x gives us x = -8. Hence, the x-intercept is \(-8, 0\). This means our line touches the x-axis at the point -8.
y-intercept
The y-intercept of a linear equation is the point where the line crosses the y-axis. To find the y-intercept, you need to set the value of x to zero in the equation, and then solve for y. Using the same example \-x + 4y = 8\, we set x = 0. This gives us: \-0 + 4y = 8, simplifying down to 4y = 8. Solving for y, we find y = 2, so the y-intercept is \(0, 2\). This tells us our line crosses the y-axis at the point 2.
coordinate plane
The coordinate plane is a two-dimensional space formed by two perpendicular lines called axes. These axes are the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is identified by an ordered pair (x, y), corresponding to the distances from the point to the vertical and horizontal axes.
To effectively plot points and understand linear equations, familiarity with the coordinate plane is essential.
A few tips:
  • The origin (0, 0) is where the x and y-axes intersect.
  • Positive x values are to the right of the origin; negative x values are to the left.
  • Positive y values are above the origin; negative y values are below.
plotting points
Plotting points involves placing points on the coordinate plane based on their x and y values. For our linear equation, we found two important points: the x-intercept \(-8, 0\) and the y-intercept \(0, 2\).
Here's how to plot them:
  • Start by locating the x-intercept \(-8, 0\). Move 8 units left of the origin along the x-axis and place a point.
  • Next, plot the y-intercept \(0, 2\). Move 2 units up from the origin along the y-axis and place another point.
  • Once both points are plotted, draw a straight line passing through them. This line represents the graph of the equation \(-x + 4y = 8\).
This process can be used to graph any linear equation by determining and plotting the intercepts.

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