Problem 64 - Introductory and Intermediate Algebra for College Students 4th

Question

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-x+3$$

Step-by-Step Solution

Answer

The graph of the given equation \(y = -x + 3\) is a straight line with a negative slope, intersecting the y-axis at the point (0,3). The table of values contains the following pairs (0,3), (1,2), (2,1), (3,0), and (4,-1), all of which lies on the line.

Step 1: Start with the given equation

The given linear equation is \(y = -x + 3\). This equation is already in slope intercept form, which is \(y = mx + b\), where m is the slope and b is the y-intercept.

Step 2: Identify the slope and y-intercept

The slope, m, is the coefficient of x, which is -1 in this context. The y-intercept, b, is the constant, which is 3 in this equation. This indicates the line will cross the y-axis at the point (0, 3). The negative slope suggests the line will fall as it moves to the right.

Step 3: Construct a table of values

Begin with x = 0, substitute x into the equation to calculate y, then fill in the table with at least five pairs of values. For example, with x values of 0, 1, 2, 3, and 4 we calculate the following y values: 3, 2, 1, 0, and -1, respectively.

Step 4: Plot the points and draw the line

Plot the points from the generated table of values on the Cartesian plane. The points should align in a straight line. Draw a straight line through the points, ensuring it passes through the y-intercept (0,3). Your line should slant downward since the slope is negative.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a line in a straightforward manner. It’s written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This format is particularly useful because it immediately gives us two key pieces of information about the line: the slope and the point where the line crosses the y-axis. Knowing the slope can help you understand the direction and steepness of the line, while the y-intercept tells you exactly where the line sits at the y-axis. When working with linear equations, using the slope-intercept form can make graphing much simpler and quicker.

To solve equations in this form, identify the values of \(m\) and \(b\) from the equation. These constants will be crucial for graphing or creating a table of values.

Slope of a Line

The slope of a line is a measure of its steepness or tilt. In the slope-intercept form equation \(y = mx + b\), the slope is represented by \(m\). It shows how much \(y\) changes for a change in \(x\). A positive slope means the line rises to the right, whereas a negative slope means the line falls to the right.

In the equation \(y = -x + 3\), the slope \(m\) is \(-1\). This means for every 1 unit increase in \(x\), \(y\) decreases by 1 unit. Recognizing the slope allows you to predict the behavior of the line. In real-world contexts, slope could indicate rates of change, such as speed or growth rate.

• Positive Slope: Lines that ascend as we move from left to right.
• Negative Slope: Lines that descend as we move from left to right.
• Zero Slope: Horizontal lines, meaning no change in \(y\).
• Undefined Slope: Vertical lines, where \(x\) does not change.

Y-Intercept

The y-intercept is the point where a line crosses the y-axis. In slope-intercept form, \(y = mx + b\), the \(b\) value is the y-intercept. This value is essential as it provides a starting point for graphing a line. You can quickly mark this point on the y-axis and use it, along with the slope, to draw the line.

For the equation \(y = -x + 3\), the y-intercept is 3. This informs us that the line crosses the y-axis at the point \((0, 3)\). By plotting this point, we denote where the line begins its path across the graph. Understanding y-intercepts can also give insights into real-world scenarios, such as fixed starting points in various contexts like finance or motion.

Table of Values

A table of values is a helpful tool to illustrate how changes in \(x\) affect \(y\) in a given equation. You select different \(x\) values, substitute them into the linear equation, and compute the corresponding \(y\) values. This creates ordered pairs \((x, y)\) that can be used for graphing.

Creating a table of values for the equation \(y = -x + 3\) involves choosing a series of \(x\) values and calculating their respective \(y\) values. For instance, choosing \(x\) values of 0, 1, 2, 3, and 4 yields \(y\) values of 3, 2, 1, 0, and -1. This gives us the points \((0,3), (1,2), (2,1), (3,0), (4,-1)\).

Once plotted, these points form a straight line that can be extended to show the entire linear relationship. Using a table of values simplifies the process of visualizing linear equations and sketching precise graphs. It can also serve as a means of checking your work when ensuring graph accuracy.