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Chapter 3: Problem 31

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ y=2.5 $$

Short Answer

Expert verified
The x-intercept does not exist; the y-intercept is (0, 2.5).

Step by step solution

01

Finding the x-intercept

To find the x-intercept, set y to 0 and solve for x. However, in the equation \( y = 2.5 \), y is always 2.5 and does not depend on x. Therefore, there is no x-intercept.
02

Finding the y-intercept

To find the y-intercept, substitute x = 0 into the equation. Since the equation is y = 2.5, when x = 0, y = 2.5. Thus, the y-intercept is (0, 2.5).

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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 graph is the point where the graph crosses the x-axis. At this point, the value of y is zero. To find the x-intercept for any equation, you need to set y to 0 and solve for x.

In the original exercise provided, we have the equation:
\[ y = 2.5 \]
Here, y is always 2.5, regardless of the value of x. This means the graph does not cross the x-axis, as y never equals 0. Therefore, this equation does not have an x-intercept.
y-intercept
The y-intercept is the point where the graph crosses the y-axis. At this point, the value of x is zero. To determine the y-intercept for a given equation, you simply set x to 0 and solve for y.

For the given equation:
\[ y = 2.5 \]
When x is 0, the value of y is still 2.5. Thus, the y-intercept is the point \((0, 2.5)\). This means that the graph touches the y-axis at 2.5.
graphing linear equations
Graphing linear equations involves plotting points that satisfy the equation and then drawing a straight line through these points. Let鈥檚 understand this by looking at a simple linear equation, such as the one given in the exercise:
\[ y = 2.5 \]
Here, y is constantly equal to 2.5, irrespective of the x value. This means that every point on the graph has a y-coordinate of 2.5. To graph it:
  • Choose different values for x. It can be any number.
  • Plot the points (x, 2.5) on the graph.
  • Draw a horizontal line through all these points.

So, for this specific equation, the graph will be a horizontal line passing through y = 2.5.

Remember:
  • If the equation is in the form of y = mx + b, the graph will be a straight line.
  • The slope (m) determines the line鈥檚 steepness and direction.
  • The y-intercept (b) tells where the line crosses the y-axis.
Understanding these basics will help you effectively graph any linear equation.

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Most popular questions from this chapter

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Passes through (2,-3) parallel to \(3 x-4 y=5\)

Determine whether each ordered pair is a solution of the given equation. \(x+y=8 ; \quad(0,8)\)

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ 4 x-5 y=0 $$

Write an equation of the line satisfying the given conditions. Give the final answer in slope-intercept form. (Hint: Recall the relationships among slopes of parallel and perpendicular lines.) Perpendicular to \(x-2 y=7\) \(y\) -intercept (0,-3)

Each table of values gives several points that lie on a line. (a) Use any two of the ordered pairs to find the slope of the line. (b) What is the x-intercept of the line? The y-intercept? (c) Graph the line. $$ \begin{array}{r|r} \hline x & y \\ \hline-4 & 0 \\ \hline-2 & 2 \\ \hline 0 & 4 \\ \hline 1 & 5 \end{array} $$
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