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Chapter 4: Problem 22

Put each equation into slope-intercept form, if possible, and graph. $$5=x+2$$

Short Answer

Expert verified
The given equation is \(5 = x + 2\). We rewrite it in the slope-intercept form as \(y = -x + 3\). By identifying the slope (m = -1) and the y-intercept (b = 3), we can plot the graph. Start by plotting the y-intercept (0, 3). Then, using the slope, find another point on the line - in this case, (1, 2). Draw a line passing through both points to obtain the graph of \(y = -x + 3\).

Step by step solution

01

Convert the given equation to slope-intercept form

Given the equation \(5 = x + 2\), first, we want to solve for y. In this equation, we don't have a y term. We can rewrite the equation as \(5 = x + 2 + y - y\). Now let's move ahead: \[y = -x + 5 - 2\] \[y = -x + 3\] The equation is now in the slope-intercept form. We can identify that the slope, m, is -1 and the y-intercept, b, is 3.
02

Plot the y-intercept

Begin by plotting the y-intercept (b = 3) on the graph. This point is where the line intersects the y-axis, so its coordinates are (0, 3). Place a point at this coordinate.
03

Use the slope to find another point

Now, use the slope (m = -1) to find another point on the line. Slope (m) can be defined as the change in y divided by the change in x, or: \[m = \frac{change\ in\ y}{change\ in\ x}\] In this case, m is -1, so we can rewrite it as: \[-1 = \frac{change\ in\ y}{change\ in\ x}\] Choose an arbitrary change in x (e.g., 1) and solve for the change in y: \[-1 = \frac{change\ in\ y}{1}\] \[-1 = change\ in\ y\] Starting from the y-intercept (0,3), we can go 1 unit right along the x-axis (change in x = 1) and 1 unit down (change in y = -1) to find another point on the line. Now, we have two points: (0, 3) and (1, 2).
04

Graph the line

From the two points (0, 3) and (1, 2), draw a straight line passing through both points. This line represents the graph of the equation \(y = -x + 3\).

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations involves plotting points on a coordinate plane to represent equations of the form \(y = mx + b\), known as the slope-intercept form. This form makes it easy to understand and visualize linear relationships. To graph such an equation:
  • Identify the slope \(m\) and the y-intercept \(b\).
  • Begin by plotting the y-intercept on the y-axis, which is the point \((0, b)\).
  • Use the slope to determine the direction and steepness of the line.
Once you plot at least two points, like in the example equations given, you can draw a line through the points to represent the equation graphically. A major advantage of the slope-intercept form is that with just the slope and y-intercept, you can immediately predict how the line will look on the graph. Thus, mastering this process helps in quickly visualizing complex data.
Slope of a Line
The slope of a line, represented by \(m\) in the equation \(y = mx + b\), indicates both the direction and steepness of the line. It is calculated as the 'rise' over the 'run', meaning the change in y divided by the change in x:
  • If \(m > 0\): the line rises as it moves from left to right, showing a positive slope.
  • If \(m < 0\): the line falls as it moves from left to right, showing a negative slope.
  • If \(m = 0\): the line is horizontal, indicating no change as x varies.
In the example \(y = -x + 3\), the slope is \(-1\). This tells us that for every unit increase in x, y decreases by one unit. Slope is crucial because it tells us whether values are increasing, decreasing, or remaining constant, and helps predict how changes in variables relate to one another.
Y-intercept
The y-intercept is the point where a line intersects the y-axis, represented by \(b\) in the slope-intercept form \(y = mx + b\). It tells us the value of y when x is zero. Graphically, this is the starting point for plotting the line of a linear equation. In our example, the y-intercept is 3, meaning the point (0,3) lies on the line.
  • Find it by setting \(x = 0\) in the equation.
  • This is the first point to plot when graphing.
  • Gives the initial value of the variable y as x increases or decreases.
Understanding the y-intercept is important, as it provides a quick reference to the entire line's position on the graph. It anchors the line鈥檚 position and sets the context for using the slope to find further coordinates to delineate the line's path.

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Most popular questions from this chapter

Jenelle earns \(\$ 7.50\) per hour at her part-time job. Her total earnings, \(E\) (in dollars), for working \(t\) hr can be defined by the function $$ E(t)=7.50 t $$ a) Find \(E(10),\) and explain what this means in the context of the problem. b) Find \(E(15),\) and explain what this means in the context of the problem. c) Find \(t\) so that \(E(t)=210,\) and explain what this means in the context of the problem.

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{1}{x+8}$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=3 x+3$$

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(y\) -intercept \((0,6)\) and \(m=7\)

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(5,-1),(7,3)\\\&L_{2}:(-6,0),(4,5)\end{aligned}$$
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