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Chapter 5: Problem 21

Graph each equation by finding the intercepts and at least ne ether point. $$x=5$$

Short Answer

Expert verified
The equation \(x=5\) is a vertical line that passes through the x-intercept (5, 0) and does not have a y-intercept. An additional point on the graph is (5, 2). To graph the equation, plot the points (5, 0) and (5, 2), and draw a vertical line passing through these points.

Step by step solution

01

Determine the x-intercept

To find the x-intercept, we substitue \(y = 0\) into the given equation and solve for \(x\). Given equation: \(x = 5\) Since the equation already has the \(x\) value, we don't need to substitute \(y\). x-intercept: (5, 0)
02

Determine the y-intercept

For this given equation, there is no y-intercept because the equation is a vertical line and does not cross the y-axis. However, we can confirm this by attempting to solve for the y-intercept. To find the y-intercept, substitute \(x = 0\) into the given equation. \(0 = 5\) Since this equation is not possible, there is no y-intercept for this given equation.
03

Find another point on the graph

To find another point on the graph, we can choose any value of \(y\), and since the equation is \(x = 5\), the \(x\) value will remain constant. Let's choose \(y = 2\). Point: (5, 2)
04

Graph the equation

Using the x-intercept and the additional point found in the previous steps, we can now graph the equation \(x = 5\). 1. Plot the x-intercept (5, 0) on the graph. 2. Plot the additional point (5, 2) on the graph. 3. Draw a vertical line that passes through both points. This line represents the equation \(x = 5\). Now you have successfully graphed the equation \(x = 5\) by finding its intercepts and at least one other point.

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Key Concepts

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

Finding Intercepts
Finding intercepts is a foundational skill in graphing equations. Intercepts are points where the graph crosses the axes. For many equations, you need to find both the x-intercept and the y-intercept.
  • To find the x-intercept, set the value of \( y \) in the equation to zero and solve for \( x \).
  • To find the y-intercept, set the value of \( x \) in the equation to zero and solve for \( y \).
In the case of the equation \( x = 5 \), the x-intercept is straightforward because the equation already gives the x value. Therefore, the x-intercept is \((5, 0)\).
For the y-intercept, we substitute \( x = 0 \) into the equation, which doesn't work here, showing no y-intercept. This is typical for vertical line equations that don't cross the y-axis.
Graphing Equations
Graphing equations involves plotting points and drawing a line or curve through them. It helps to visualize relationships and understand equations better. When graphing, two key components are required: points (often intercepts) and the shape of the line.
  • Start by plotting whatever intercepts you have found.
  • Find at least one other point to ensure you draw the line correctly.
For the equation \( x = 5 \), you've got an x-intercept at \( (5, 0) \) and another point at \( (5, 2) \). Now, you can draw the line passing through these points, extending in both the positive and negative y-directions.
Once drawn, you'll see a clear vertical line indicating the equation's behavior.
Vertical Line Equations
Vertical line equations are a special type of equation where \( x \) is equal to a constant, such as \( x = 5 \). These lines run parallel to the y-axis and never cross it.
  • They have the same x-coordinate for all points on the line.
  • They do not have a y-intercept because they do not cross the y-axis.
A vertical line equation like \( x = 5 \) maintains a fixed x-coordinate regardless of the y-value. This is why the graph is a straight vertical line at \( x = 5 \). Whether \( y \) is 2, 100, or \(-50\), as long as \( x = 5 \), the point is on the line.
This straightforward nature makes vertical line equations easy to plot but unique because of their absence of a y-intercept.

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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. \(y=-x+11 ;(-10,-8) ;\) slope-intercept form

Solve each systen \(\begin{aligned} 4 a+2 b &=-11 \\\\-8 a-3 c &=-7 \\ b+2 c &=1 \end{aligned}\)

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. Write an equation of the line perpendicular to \(x=7\) containing \((6,0)\)

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slopeintercept form or in standard form, as indicated. $$x-4 y=9 ;(5,3) ; \text { standard form }$$

(4.6) Determine if each pair of lines is parallel, perpendicular or neither. $$\begin{aligned} &y=6 x-7\\\ &4 x+y=9 \end{aligned}$$
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