Problem 76 - Introductory Algebra for College Students

Question

graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{5}{2} x+1$$

Step-by-Step Solution

Answer

The graph of the equation \(y = -\frac{5}{2}x + 1\) starts at the y intercept (0,1), with a slope of -5/2. The five solutions to the equation are (0,1), (2,-4), (4,-9), (6,-14), (8,-19).

Step 1: Identify the slope and y-intercept

The equation is in slope-intercept form \(y = mx + b\). In this equation, \(m\) is the slope and \(b\) is the y-intercept. For our equation \(y = -\frac{5}{2}x + 1\), the slope \(m\) is \(-\frac{5}{2}\) and the y-intercept \(b\) is 1.

Step 2: Plot the y-intercept

Start the graph by plotting the y-intercept, which is at point (0,1). It indicates that when \(x = 0\), \(y = 1\). So, plot a point at (0,1) on the graph.

Step 3: Use the slope to find the next points

The slope \(-\frac{5}{2}\) means that for every 2 units move to the right along the x-axis (positive direction), we move 5 units down along the y-axis (negative direction). This will give us the next point (2,-4). Continue this process to find more points: (4,-9), (6,-14), (8,-19).

Step 4: Plot the points and draw the line

Plot the points (2,-4), (4,-9), (6,-14), (8,-19) on the graph. Draw a straight line passing through all of these points. This line is the graph of the equation \(y = -\frac{5}{2}x + 1\) .

Step 5: Write down the solution

The five solutions of the equation \(y = -\frac{5}{2}x + 1\) are (0,1), (2,-4), (4,-9), (6,-14), (8,-19). Each of these ordered pairs is a solution because they make the equation true.

Key Concepts

Graphing Linear Equations

Graphing linear equations is a way to visualize how two variables relate to each other. In a coordinate system, the equation forms a straight line. To graph a linear equation, you need several key points that lie on this line.
Start by creating a table of values. Pick different values for one variable, usually the independent variable, and solve for the other. These values form points on a graph.

• Select convenient values, including zero, for precision and ease.
• Plot these points on the graph.

Once the points are plotted, draw a line through them. This line represents all possible solutions to the equation.

Slope-Intercept Form

The slope-intercept form is one of the simplest ways to write a linear equation. It's structured as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
This format makes it easy to understand and graph an equation quickly:

• \(m\), the slope, indicates the steepness or direction of the line.
• \(b\), the y-intercept, tells you where the line crosses the y-axis.

Using this form helps you graph equations without needing additional calculations. You can directly plot the y-intercept and use the slope to determine other points.

Finding Solutions of Equations

Finding solutions to a linear equation means discovering points that make the equation true. For each solution, both sides of the equation are equal when substituted with the values of \(x\) and \(y\).
Here's how you can find solutions:

• Select an \(x\) value.
• Calculate the corresponding \(y\) using the equation.

Repeat this process to find multiple points. These solutions can be plotted on a graph to form the line of the equation.
The more points you calculate, the more accurately you can plot the line.

Y-intercept and Slope

The y-intercept and slope are crucial elements of a linear equation in slope-intercept form. The y-intercept \(b\) is the point where the line crosses the y-axis, symbolized as \((0, b)\). It shows the value of \(y\) when \(x = 0\).
The slope \(m\) describes how steep the line is and its direction of increase or decrease. It is often written as a ratio \(\frac{rise}{run}\), showing vertical change over horizontal change.

• A positive slope means the line goes up as you move right.
• A negative slope means the line goes down as you move right.

Understanding these concepts helps in graphing and interpreting how variables relate to each other in an equation.