Problem 73 - Introductory Algebra for College Students

Question

Read Exercise \(72 .\) Then use a graphing utility to solve the systems. $$\left\\{\begin{array}{l}y=2 x+2 \\ y=-2 x+6\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is the intersection point of the two lines on the graph, which in this case is the point \( (1, 4) \).

Step 1: Plot the equations

First, plot both lines on the graph. The first line \(y=2x+2\) is a linear function with a slope of 2 and y-intercept 2. The second line \(y=-2x+6\) is also a linear function but with a slope of -2 and y-intercept 6.

Step 2: Find the intersection point

Next, observe where the two lines intersect, this will be the solution to the system. This is because the solution of a system of linear equations is the point where the graphs of the two equations meet.

Step 3: Extract the coordinates

Finally identify the coordinates of the intersection point. These coordinates are the solutions to the system of equations, representing the x and y values that satisfy both equations.

Key Concepts

Graphing Utility

When solving systems of linear equations, a graphing utility is an invaluable tool that aids visualization and increases accuracy. With the click of a button, you can plot graphs of equations and pinpoint their points of intersection with precision.

A graphing utility is essentially software or a feature within a calculator that allows you to input equations and draw their graphs on a coordinate system. It simplifies the process of finding where two or more equations intersect, which would represent the solution to the system. Using a graphing utility saves time and reduces the potential for human error that comes with manually sketching graphs. For instance, in Exercise 72, you can input the given linear functions and the graphing utility will show you exactly where they cross.

Linear Functions

Linear functions are at the very heart of algebra and represent relationships with a constant rate of change, which we see as straight lines on a graph. These functions have the general form of \(y=mx+b\), where \(m\) represents the slope, and \(b\) represents the y-intercept. The slope tells us how steep the line is, and the y-intercept is the point at which the line crosses the y-axis.

In the context of the exercise, both given equations are linear functions. Each equation models a straight-line graph where every point on that line is a solution to that equation. When combined, these equations form a system whose solution must satisfy both equations simultaneously. Visual understanding of linear functions is crucial for constructing and interpreting graphs and their intercepts.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \(y=mx+b\). This direct and informative format gives a quick snapshot of the graph of a line. Here, \(m\) is the slope, which indicates how many units the line moves up or down for every unit it moves right. The \(b\) is the y-intercept which shows where the line crosses the vertical y-axis.

Understanding the slope-intercept form is pivotal; it allows you to quickly sketch the basic shape of a line without needing a table of values or additional calculations. For instance, the first equation from the exercise, \(y=2x+2\), tells us straight away that the line crosses the y-axis at 2 and slopes upwards since the slope is positive. Conversely, \(y=-2x+6\) has a negative slope, meaning the line will slope downwards.

Intersection Point

The intersection point of two lines on a graph is a single point where both lines meet. This special point is the key to solving systems of linear equations because it represents the set of x and y values that are true for both equations. In simple terms, it's the 'winning ticket' that satisfies all the conditions set by the system.

Graphically identifying the intersection point is straightforward using a graphing utility. In a system such as the one in Exercise 72, you find this point where the two graphed lines intersect. The coordinates of this point are the solution to the system. For example, if lines intersect at the point (3,8), then x=3 and y=8 are the solutions that satisfy both original equations simultaneously.