We are asked to explain the similarities and differences between finding the solution of a and finding the solution of a . 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 . We will illustrate how to do this graphically. The solution for the system are all that satisfy both at the same time. Graphically, these points are where both representing the intersect. Let's use an example system.
{y=x+3y=-x+3
We start by writing each equation in this system in .
{y=1x+3y=-1x+3
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 , that is (0,3).
Finding the solution for a system of linear inequalities
We will illustrate how to solve a system of 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.
{y>x+3y<-x+3
To find the solution set, we will first graph the . These can be obtained by replacing the inequality signs with equal signs.
System of inequalities{y>x+3y<-x+3 Boundary lines ⇒ {y=x+3y=-x+3
In our example system, we will use dotted lines since the inequality signs are
> and
<. If we had used
≥ or
≤, we would have solid lines instead.
Now we use a test point to determine the region of the for each individual inequality. The overlapping region will be the . 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<-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.