3.4 Applications of Systems of Equations

3.4 Applications of Systems of Equations

By Rayne Young

Multiple Choice

A ______ is a set of two or more equations that have the same variables.

• solution of a system
• elimination method
• system of equations
• table

Multiple Choice

The ordered pair that satisfies all equations in the system is the

• system of equations
• function
• graph
• solution of a system

Multiple Choice

When solving a system of equations algebraically, which statement results in infinite solutions?

• 4 = 7

• 2 = 2

• -3 = 3

• all of the above

Multiple Choice

When solving a system of equations, which statement results in no solution?

• 4 = 7

• -2 = 2

• 0 = 3

• all of the above

Ways to Solve:

Can you follow each method?

Given two equations & the method, can you determine proper steps in solving?

STEPS in Solving

Elimination

Given x = 2y + 1 and 2x + 3y = 12.

1) Plug ​in for x

2(2y + 1) + 3y = 12

2) Simplify​

4y + 2 + 3y = 12

7y + 2 = 12

3) Solve​ for y

4) Plug back in & solve for x​

Substitution

​Given -3x + 4y = 24 and 2x + 3y = 12.

1) SAME coefficient w/ OPPOSITE signs (multiply x or y by opposite coefficient)

​ 2(-3x + 4y = 24)

3 (2x + 3y = 12)​

2) Add down

​ 0x + 17y = 84

3) Solve​ for y

4) Plug back in & solve for x​

​-6x+8y=48

6x+9y=24​

Multiple Select

A system of equations is shown below.

$$4x-5y=-10$$  

$$2x-7y\ =4$$  

Which operations on the system of equations will isolate the y quantity?

(There are two correct options.)

• Cancel the y's by multiplying the top by 2 & bottom by 4.

• Cancel the x's by multiplying the top by 2 & bottom by 4 (make one negative).

• Cancel the x's by multiplying the top by 4 & bottom by 2.

• Cancel the y's by multiplying the top by 7 & bottom by 5 (make one negative).

• Cancel the y's by multiplying the top by 5 & bottom by 7.

Multiple Choice

The substitution method will be used to solve this system of equations.

$$x=7-2y$$  

$$2x-7y\ =3$$  

Which equation would lead to a correct solution with this method?

• $$\left(7-2y\right)+2y=7$$  

• $$\left(7+2y\right)+2y=7$$  

• $$2\left(7-2y\right)-7y=3$$  

• $$2\left(7+2y\right)-7y=3$$  

Writing Equations: Can you set it up?

Given a word problem, can you write the 2 equations to model the situation?

Setting Up a System

Don't forget to define your variables!

Then, use the info from the word problem to pair variables with the numbers that go with them.

Matt and Ming are selling fruit for a school fundraiser. Customers can buy small boxes of oranges and large boxes of oranges. Matt sold 3 small boxes and 14 large boxes for a total of $203. Ming sold 11 small boxes and 11 large boxes for a total of $220. Find the cost each of one small box and one large box of oranges.

​

x = cost of small boxes

y = cost of large boxes

​

Matt: 3x + 14y = 203

Ming: 11x + 11y = ​220

Multiple Choice

Some students want to order shirts with their school logo. One company charges $9.65 per shirt plus a setup fee of $43. Another company charges $8.40 per shirt plus a $58 fee. Which equation represents the number of shirts when both companies charge the same amount? 

• y = 9.65 + x
  y = 8.40 + x
• y = 9.65x + 43
  y = 8.40x + 58
• y =9.65x
  y = 8.40x
• y = 9.65x - 43
  y = 8.40x - 58

Multiple Choice

On Monday Mr. Beignet bought 10 coffees and 5 doughnuts for his office at the cost of $16.50. On Tuesday he bought 5 cups of coffee and 10 doughnuts for a total of $14.25. Which equations could be used to determine the cost of each item?

• 10c + 5d = 14.25

  5c + 10d = 16.50

• 10c + 5d = 16.50

  5c + 10d = 14.25

• c + d = 10

  5c + 10d = 16.50

• c + d = 5

  5c + 10d = 16.50

Multiple Choice

Michelangelo decided to order Pizza Hut last night. He purchased 3 pizzas and 2 orders of breadsticks for a total of $29.50. Donatello ordered 2 pizzas and 3 orders of breadsticks last Sunday for a total of $23. Set up a system. Let "p" represent the price per pizza and "b" represent the price of an order of breaksticks.

• 3p + 2b = 29.50
  2p + 3b = 23
• 3p + 2b = 23
  2p + 3b = 29.50
• 3b + 2p = 29.50
  2b + 3p = 23
• 5bp = 29.50
  5bp = 23

Interpret Solutions:

What do the #s mean?

Given the solution in context, can you explain what each number represents?

Interpreting Solutions

Give context to your answers!

Matt and Ming are selling fruit for a school fundraiser. Customers can buy small boxes of oranges and large boxes of oranges. Matt sold 3 small boxes and 14 large boxes for a total of $203. Ming sold 11 small boxes and 11 large boxes for a total of $220. Find the cost each of one small box and one large box of oranges.

​

x = cost of small boxes

y = cost of large boxes​

Matt: 3x + 14y = 203

Ming: 11x + 11y = ​220

​Small boxes (x) cost $7.

Large boxes (y) cost $13.​

​After solving, you find get (7,13), or x=7 & y=13 . What do these numbers mean?​

Multiple Choice

Alexandra finds that she can give 3 haircuts and 2 hair dyes in 315 minutes. Giving 2 haircuts and 4 hair dyes takes 450 minutes.  After solving the system below, Alexandra gets a solution of (45,90)

3x + 2y = 315

2x + 4y = 450

What does the solution (45,90) mean?

• Hair dyes take Alexandra 45 minutes & haircuts take 90 minutes.

• Haircuts take Alexandra 45 minutes & hair dyes take 90 minutes.

• Haircuts cost $45 & hair dyes cost $90.

• Hair dyes cost $45 & haircuts cost $90.

Multiple Choice

Caroline is considers 2 video game rental plans. Plan A can be modeled with the equation  $$C=2n$$ , and Plan B is modeled with the equation  $$C=n+6$$ , where C represents the cost in dollars & n represents the number of games rented each month. The solution to this situation is $$n=6$$ & $$C=12$$ . What do these numbers mean?

• Caroline can order 12 games from Plan A & 6 games from Plan B.

• If Caroline orders 12 games each month, both plans will cost $6.

• Caroline can order 6 games from Plan A & 12 games from Plan B.

• If Caroline orders 6 games each month, both plans will cost $12.