applying systems of equations

1. Applying Systems
of Equations
AN EXAMPLE TO HELP YOU REVIEW MODULE 6 LESSON 4…
M.WILLATT

2.  The North Carolina State Fair is a popular field trip destination. This
year the Math I class at Willatt High School and the Math II class at
Midgette High School both planned trips there.
 Willatt High School rented and filled 8 vans and 8 buses with 240
students. Midgette High School rented and filled 4 vans and 1 bus with
54 students.
 Every van had the same number of students in it as did the buses.
 Find the number of students in each van and in each bus.

3.  First, we should pick our variables. We already know how many of
each vehicle we need, but we don’t know how many students fit in each
vehicle. Since that is our unknown, that should be what our variables
represent:
x = number of students per van
y = number of students per bus

4.  Second, we can set up our equation to represent Willatt H.S.:
• W.H.S. rented and filled 8 vans and 8 buses with 240 students.
• So to find the total amount of their students in vans, we can multiply the number
of vans times the number of students that fill a van: 8x
• Likewise to find the total on buses, we can multiply the number of buses times
the number of students that fill a bus: 8y
• When we add bus students & van students, the total is 240:
8x+8y=240

5.  Third, we can set up our equation to represent Midgette H.S.:
• M.H.S. rented and filled 4 vans and 1 buses with 54 students.
• Again, the total amount of their students in vans: 4x
• Likewise the total amount of their students on buses: 1y
• When we add bus students & van students, the total is 54:
4x+1y=54

6. Now that we have our system of equations…
8x+8y=240
4x+1y=54
Remember from the Lesson 4 Notes, that you have 4 ways to solve the system
of equations:
Tables Substitution Graphing Elimination

7. Table
It will be easier if we solve the equations for y:
8x+8y=240  8y=-8x+240  y=-x+30
4x+1y=54  1y=-4x+54
Then we can plug in x values & look for a shared (x,y):
y=-x+30 y=-4x+54
x y
6 24
7 23
8 22
9 21
x y
6 30
7 26
8 22
9 18

8. Graph
• Once we graph, we’re
looking for intersection
between our equations.
• Since (8,22) is included in
both lines, it is a solution
for both… thus it is a
solution to the system of
equations.

9. Substitution
To solve by substitution, we
need to get a variable alone in
one equation. Let’s try y :
8x+8y=240
8y=-8x+240
y=-x+30
Lastly, sub in the x value to
find y’s value:
y=-(8)+30
y=22
Then substitute what y
equals in place of the y in
the other equation and
solve for x:
4x+1y=54
4x+1(-x+30)=54
4x-x+30=54
3x+30=54
3x=24
3x/3=24/3
x=8

10. Elimination
When the equations are added, one variable needs to be
eliminated. To do this, you can multiply the entire equation(s) by
a number so that a variable will cancel. I know 4x*-2 equals -8x so
when it adds to 8x, they’ll cancel.
8x+8y=240  leave alone 8x+8y=240
4x+1y=54  times negative 2  + -8x-2y=-108
6y = 132
Then plug your solution for y into 6y/6=132/6
either equation: y=22
4x+1(22)=54  4x=32 x=8

11. In the end, each van held 8 students and each
bus held 22 students…
and more importantly, everyone enjoyed their
time at the fair!