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!