Solving System of Equations/Writing Equations

Solving Systems of Equations

Methods We Have Used to Solve Systems of Equations

• Graphing

• Elimination

• Substitution

Multiple Choice

^{What is the solution?} 

• 1
• -2
• (1, 2)
• (1, -1)

Multiple Choice

How many solutions will this system have? 

• No solution
• One Solution
• I Don't Know
• Infinitely Many Solutions

Multiple Choice

When you graph the exact same equation twice,

• you will have no solution. 
• you will have one solution.
• you will have infinite solutions. 
• you will graph a giraffe. 

Substitution

• Used when we we have a single variable isolated.

• The coefficient of the isolated variable should be 1.

• Replace the variable in 1 equation, with the isolated variable from the other equation.

• Find the other solution by replacing the unknown variable with the newly found variable.

Multiple Choice

What is the best method to use?

2x − 3y = −1

y =x − 1

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

What is the best method do you use?

y = 6x − 11

−2x − 3y = −7

• Graphing

• Substitution because a variable is defined

• Elimination because both equations are in standard form.

Elimination

• Best when the equation is in Standard Form

• Look for matching coefficients and variables that OPPOSITE signs

• If needed you can multiply by a constant to make variables match and be opposite

• 
  

Multiple Choice

What is the best method to use?

−4x − 2y = −12

4x + 8y = −24

• Graphing

• Substitution because a variable is defined

• Elimination because the equations are in standard form.

Multiple Choice

What variable do you eliminate?

4x + 8y = 20

−4x + 2y = −30

• X because they have opposite signs

• Y because they have opposite signs

Writing Systems of Equations

• Identify the variables

• Look for slopes and starting (y-intercepts)

• Write the equation

• Solve the equation using your preferred method

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

A large pizza at Palanzio’s Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese pizza at Guido’s Pizza is $7.30 plus $0.65 for each topping. Which system of equations could be used to find the number of toppings when both companies cost the same amount? 

• y = 6.80 + .65x
  y=7.30+.90x
• x + y = 6.80
  x + y = 7.30
• y = 6.80+.90x
  y = 7.30 + .65x
• y + .90x = 6.80
  y + .65x = 7.30

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

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.