A paper company needs to ship paper to a large printing business. The paper will be shipped in small boxes and large boxes. The volume of each small box is 6 cubic feet and the volume of each large box is 12 cubic feet. A total of 27 boxes of paper were shipped with a combined volume of 252 cubic feet. Graphically solve a system of equations in order to determine the number of small boxes shipped, x, and the number of large boxes shipped, y.
You need to create two equations by translating the English above into numbers and other mathematical symbols. Then, you need to graph each equation on the same grid. The point (x,y) where the lines intersect gives you the x and y that satisfy the conditions of the problem.
The problem already identifies your unknowns. Usually, you have to do that on your own.
x: # of small boxes shipped
y: # of large boxes shipped
What else do we know about x and y? We know their sizes. We won't use the units (cubic feet) in our equations. Since the total volume shipped is equal to the volume of all the small boxes plus the volume of all the large boxes, we have the following equation:
252 = 6x + 12y
We know that the total number of boxes is 27, leading to the following equation:
x + y = 27
Plot each of these, perhaps by making a table, perhaps by using a graphing calculator. To create a table, choose various (probably three different ones) values of and figure out what the y value must be to make the equation above true. For instance, if x = 0, then y = 27. If x = 15, y = 12. If x = 10, y = 17. That gives you three points [(0,27); (15,12); (10,17)] to plot. Connect the points and you'll have one line.
NOTE: You can simplify the first equation (252 = 6x + 12y) by dividing every term by 6 to get an equivalent line: 42 = x + 2y
This will be easier to plot if you are doing it by hand on graph paper.
I'll let you find three points for the second equation. Plot the points, connect to form a line, find the point where the two lines intersect for that is your solution!
