Systems of Equations

​Warm Up

​Solve each equation for "y"

​-2x - y = 1
-6x = 3y +3

Systems of Equations

● A system of equations is a set of two or

more equations using the same variables.

● The solution to the system is the intersection point

that satisfies all of the equations. The point will be an ordered pair.

● If there is a solution to the system of equation

the equations are called simultaneous equations.

Solutions to Systems of Equations

● Intersecting Lines – the point where the lines

intersect is THE SOLUTION

● Parallel Lines – since they never cross, there is

NO SOLUTION

*same slope, different y-intercept*

● Coinciding (overlapping) Lines – Since the lines

are on top of each other there are INFINITE
SOLUTIONS

Graphing

Systems of Equations

Graphing

Graph the line $$y=-\frac{5}{2}x-1$$

Graphing

Graph the line $$y=\frac{1}{2}x+5$$

Graphing

Systems of Equations

​What is the solution?

Graphing

Systems of Equations

Graphing

Graph the line $$y=-\frac{1}{2}x+2$$

Graphing

Graph the line $$y=\frac{1}{3}x+2$$

Graphing

Systems of Equations

​What is the solution?

Graphing

Systems of Equations

Applications of Systems of Equations

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, which statement would be considered infinite solutions?

• 4 = 7

• 2 = 2

• -3 = 3

• all of the above

Multiple Choice

When solving a system of equations, which statement would be considered

no solution?

• 4 = 7

• -2 = 2

• 0 = 3

• all of the above

Writing a System of Equations

Setting Up a System

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 Matt: 3x + 14y = 203

y = cost of large boxes Ming: 11x + 11y = ​220

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.

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

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.