To solve a system of linear equations graphically means graphing the lines and identifying the point of intersection.For example, the following system,
The point where the lines intersect is the solution to the system.
The lines appear to intersect at (1.5,2.5). Thus, this is the solution to the system.
In a football game, the home team, the Mortal Wombats, defeated the Fearless Seagulls by 13 points. The total score for both teams was 41. What was the final score?
Now, we can identify the point of intersection.
The point of intersection is (14,27). This means, the Wombats scored 27 points and the Seagulls scored 14.
The sum of two numbers is 17. One of the numbers is two more than three times the other number. Write a system that represents the given relationships. Then, find the numbers using substitution.
To graph the inequalities, begin with the boundary lines. The inequality y<-x+7 has the boundary line y=-x+7. Since the inequality is strict, the line should be dashed and, in this case, the shaded region lies below the line.
Similarly, y<-0.5x+5 has the boundary line y=-0.5x+5. Since the inequality is not strict, the line is solid. In addition, the region to be shaded lies below the line. This inequality will be graphed on the same coordinate plane.
Notice that the individual solution sets overlap in a portion of the plane. This overlapping region is the solution set of the system. All the points in this region satisfy both inequalities simultaneously. In the next graph, only the common region is plotted.
Finally, since the boundary lines in their entirety are not part of the solution set, crop them only to show the edges of the overlapping region.
Marco's mother asks him to buy burritos and tacos from the restaurant near their house. She gives him $90 and instructs him to get enough food so that they can feed 10 people. If burritos cost $5 each and tacos cost $3 each, how many of each can he buy?
The purple region represents the solution set that satisfies both inequalities. Since Marco can't buy a negative number of burritos or tacos we're only interested in the positive values of b and t.
Any point in this region corresponds to a combination on burritos and tacos that costs less than $90 and feeds at least 10 people. Let's look at the corners of this region.
The marked points represent minimum and maximum possibilities.
Since we don't know how hungry the guests are, or what their preferences are, Marco should buy both burritos and tacos. Let's choose a point in the middle of the region.
One possibility is that Marco can purchase 12 tacos and 8 burritos. That way, there will probably be enough food, and he'll have money left. Note that even though decimal numbers are a part of the solution set, the answer should be given in whole numbers, assuming you can't buy a part of a taco or burrito.