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Chapter 7: Problem 5

In Exercises 5-12, solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}x+y=6 \\ x-y=2\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of equations is (2,4).

Step by step solution

01

Translate to Slope-Intercept Form

The first step is to rearrange both equations into the y = mx + b form, which is the slope-intercept form. Here, m stands for the slope of the line and b is the y-coordinate where the line crosses the y-axis (y-intercept). So for the first equation x + y = 6, it can be rearranged to y = -x + 6. And the second equation x - y = 2 can be rearranged to y = x - 2.
02

Graph Both Equations

The next step is to graph the two equations. The slope of the first line is -1 and the y-intercept is 6, so plot the point (0,6) and then move one point down and one to the right. The slope of the second line is 1 with y-intercept -2, so plot the point (0,-2) and then move one point up and one to the right. The intersection of the two lines is the solution to the system.
03

Find Intersection Point

By observing the graph, the point of intersection can be determined. The lines represented by the equations y = -x + 6 and y = x - 2 intersect at the point (2,4). So the solution to this system of equations is (2,4). This means x=2 and y=4.
04

Check the Coordinates

Lastly, substitute x=2 and y=4 into the original equations to prove that they hold true. For x+y=6, substituting these values in gives 2+4 which is equal to 6. And for x-y=2, substituting these values gives 2-4 which is equal to -2. Both equations hold true so we can confirm that the solution is x=2, y=4.

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