Exercise 6 Page 402

We are asked to explain the similarities and differences between finding the solution of a system of inequalities and finding the solution of a system of equations. Let's recall each case first, and then draw the conclusions.

Finding the solution for a system of linear equations

There are many ways to solve a system of linear equations. We will illustrate how to do this graphically. The solution for the system are all points that satisfy both linear equations at the same time. Graphically, these points are where both lines representing the equations intersect. Let's use an example system. We start by writing each equation in this system in slope-intercept form. We will now graph each equation.

Note that the lines intersect at just one point and therefore have just one solution.

The solution for our example is the point of intersection, that is $(0,3).$

Finding the solution for a system of linear inequalities

We will illustrate how to solve a system of linear inequalities graphically. The solution for the system will be the set of all points that satisfy both linear inequalities at the same time. Let's work on an example system. To find the solution set, we will first graph the boundary lines. These can be obtained by replacing the inequality signs with equal signs. In our example system, we will use dotted lines since the inequality signs are $>$ and $<.$ If we had used $\ge$ or $\le ,$ we would have solid lines instead. Now we use a test point to determine the region of the solution set for each individual inequality. The overlapping region will be the solution set of the system. We will use the test point $(0,0)$ for simplicity. Let's test $y>x+3$ first. We arrived at a contradiction. It means that the point $(0,0)$ is not part of the solution set for $y>x+3.$ Therefore, we should shade the region which does not contain the test point. Let's now test the second inequality, $y<\text{-}x+3.$ We got a true statement for this case. This implies that the point is part of the solution set. Thus, we should shade the region containing it.

The solution set for the system is the region shown below.

Conclusion

We can summarize the similarities and differences between finding the solution of a system of inequalities and finding the solution of a system of equations.

Similarities

• The solution set for both systems is all points that satisfy the whole system at the same time.
• To find the solution set graphically, we need to graph two straight lines first.

Differences

• For the system of linear equations, graphing the two lines is enough. For the system of linear inequalities the two lines just represent the boundary lines. We still need to find the regions satisfying each inequality.
• The solution set for the system of linear equations is the points where the lines intersect. The solution set for the system of linear inequalities is the region created by the overlapping individual regions of each inequality composing the system.