Problem 80 - Introductory Algebra for College Students

Question

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}y=-\frac{1}{2} x+2 \\ y=\frac{3}{4} x+7\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the point where the graphs of the two equations intersect. The coordinates of this point will depend on the accuracy of your graphing and interpretation. For higher accuracy, it is recommended to use a graphing calculator or software.

Step 1: Graph the first equation

Start by graphing the first equation \(y=-\frac{1}{2} x+2\). This represents a straight line with a slope of -1/2 and a y-intercept of 2. Plot the y-intercept at (0,2) and then use the slope to find another point on the line. Since the slope is -1/2, for every 2 units you move to the right, move 1 unit down. Draw a line through these two points.

Step 2: Graph the second equation

Next, graph the equation \(y=\frac{3}{4} x+7\). This line has a slope of 3/4 and a y-intercept of 7. Plot the y-intercept at (0,7) and then use the slope to find another point on the line. Since the slope is 3/4, for every 4 units you move to the right, move 3 units up. Draw a line through these two points.

Step 3: Identify point of intersection

The point where the two lines intersect represents the solution to the system of equations. It is the x- and y-values that satisfy both equations simultaneously. Identify this point on the graph.

Key Concepts

graphing utility

A graphing utility is a powerful tool that aids in graphing equations and visualizing their solutions. Many graphing utilities are available as software applications or even handheld devices.
They are indispensable when dealing with complex equations or systems such as the one given in the exercise:

• They allow for quick graphing of multiple equations at once.
• You can easily and accurately find points of intersection, which are crucial when solving systems of equations.
• They provide clear visual representations of how equations relate to each other on a coordinate plane.

For linear equations like those in the exercise, a graphing utility allows you to see how each line behaves, making it easier to find their point of intersection. This is essential in identifying the solution to the system of equations. Using a graphing utility ensures accuracy and saves time, especially when hand-drawing the lines might lead to errors. By inputting the equations into the tool, you can instantly see their graphs and find their intersection with precision.

slope-intercept form

The slope-intercept form of a linear equation is a way of writing the equation so you can easily plot it on a graph. It is written as: \[ y = mx + b \]where:

• \( m \)is the slope of the line which indicates the steepness and direction of the line.
• \( b \)is the y-intercept, the point where the line crosses the y-axis.

In the exercise, the equations are already in slope-intercept form:
- The first equation \( y = -\frac{1}{2}x + 2 \) means the slope \( m \) is \(-\frac{1}{2}\) and the y-intercept \( b \) is 2.
- The second equation \( y = \frac{3}{4}x + 7 \) means the slope \( m \) is \( \frac{3}{4} \) and the y-intercept \( b \) is 7.
Knowing the slope-intercept form makes it easier to quickly identify these critical points for graphing. Starting with the y-intercept, and using the slope to determine the rise and run, we can plot the line with ease. Understanding and using this form is fundamental to handling systems of linear equations effectively.

linear equations

Linear equations are fundamental in algebra and mathematics as they represent straight lines when graphed. They are called "linear" because the graph of such equations is a straight line.
Linear equations in the format \( y = mx + b \) show a direct relationship between two variables:

• The change in the dependent variable \( y \) is consistent with changes in the independent variable \( x \).
• The slope \( m \) determines the angle and direction of the line, with positive slopes rising and negative slopes falling as you move from left to right.

The solutions to a system of linear equations are the points where the graphs of the equations intersect.
For the exercise, solving the system means finding where the two equations \( y = -\frac{1}{2}x + 2 \) and \( y = \frac{3}{4}x + 7 \) intersect. These points of intersection give the values of \( x \) and \( y \) that make both equations true.
Linear equations are a foundational concept, providing insights into the relationships between variables and serving as a basis for more complex mathematical studies.