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Chapter 4: Problem 21

Graph each equation by finding the intercepts and at least one other point. $$y=-x$$

Short Answer

Expert verified
The x-intercept and y-intercept are both at the point (0, 0). Another point on the graph is (1, -1). Plot these points on the graph and draw the line passing through them to graph the equation \(y = -x\).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set the value of y to 0 and solve for x. \[y = -x\] \[0 = -x\] \[x = 0\] The x-intercept is at the point (0, 0).
02

Find the y-intercept

To find the y-intercept, set the value of x to 0 and solve for y. \[y = -x\] \[y = -0\] \[y = 0\] The y-intercept is also at the point (0, 0).
03

Find another point on the graph

We can choose any value for x to find another point on the graph. Let's choose x = 1 to solve for y. \[y = -x\] \[y = -1\] The point (1, -1) is another point on the graph.
04

Graph the equation

Now that we have three points: (0, 0), (0, 0), and (1, -1), we can plot these points on the graph. Since the x-intercept and y-intercept are both at the origin (0, 0), the graph passes through the origin. The line also passes through the point (1, -1). Plot these points on the graph paper and draw the line that passes through them. This will give you the graph for the equation \(y = -x\).

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Key Concepts

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

x-intercept
When graphing linear equations, finding the **x-intercept** is one of the first steps you should take. The x-intercept of a line is where it crosses the x-axis. At this point, the value of y is always zero, because the point lies on the horizontal axis. To find the x-intercept from an equation like \(y = -x\), we substitute \(y = 0\) into the equation and solve for \(x\).

Here's how it works for our example:
  • Start with the equation: \(y = -x\).
  • Set \(y = 0\): \(0 = -x\).
  • Solve for \(x\): \(x = 0\).
This calculation shows that the x-intercept is at the point \((0, 0)\). This means our line crosses the x-axis at the origin. Knowing this point helps to sketch the graph accurately.
y-intercept
Finding the **y-intercept** is equally important as finding the x-intercept when plotting a linear equation. The y-intercept is where the line crosses the y-axis. At this point, the value of x is zero because the point is on the vertical axis.

To find the y-intercept from an equation like \(y = -x\), we substitute \(x = 0\) into the equation and solve for \(y\). This is how to do it:
  • Start with the equation: \(y = -x\).
  • Substitute \(x = 0\): \(y = -0\).
  • Calculate \(y\): \(y = 0\).
Thus, the y-intercept is at the point \((0, 0)\). Just like the x-intercept, our y-intercept is at the origin, and this point indicates where the line touches the y-axis, aiding us in drawing the graph.
plotting points
After finding the x-intercept and y-intercept, the next step in graphing a linear equation is **plotting points**. This involves identifying additional points that lie on the line to ensure its accurate representation. Since linear equations form straight lines, you only need a few points to draw the complete line.

Let's look at how we can find an additional point for our equation \(y = -x\):
  • Choose a value for x, such as \(x = 1\).
  • Substitute \(x = 1\) into the equation to find \(y\): \(y = -1\).
  • This gives us the point \((1, -1)\).
Now, you have three points to plot on the graph: \((0, 0)\), \((0, 0)\), and \((1, -1)\). Place these points precisely on graph paper and draw a straight line through them. This line correctly represents the linear equation \(y = -x\), with each plotted point verifying its accuracy.

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Most popular questions from this chapter

Write an equation of the line perpendicular to the given line and containing the given point. Write the answer in slope-intercept form or in standard form, as indicated. $$2 x+5 y=11 ;(4,2) ; \text { standard form }$$

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. $$y=2 x+1 ;(-2,-7) ; \text { standard form }$$

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. \(18 x-3 y=9 ;(2,-2) ;\) slope-intercept form

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. \(y=4 x+9 ;(0,2) ;\) slope-intercept form

Write the slope-intercept form of the equation of the line, if possible, given the following information. horizontal line containing \((0,-8)\)
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