Systems of Equations: Graphing, Substitution, and Elimination!

Systems of Equations: Graphing, Substitution, and Elimination!

By Shalynne Orth

System of Equations

• two or more linear equations

• solution is the intersection of the equations

Systems of Equations: Graphing

Different types of solutions

1. One solution: The solution is where the points intersect

2. No solution: The lines will never intersect so this system does not have an answer

3. Infinite solutions: The lines have the same equation.

Multiple Choice

How many solutions will this system have?

• One
• Two
• No Solution
• Infinitely Many Solutions

Multiple Choice

How many solutions will this system have? 

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

Multiple Choice

How many solutions will this system have?

• No Solution
• Infinitely Many Solutions
• No Clue
• One solution

Multiple Choice

What is the solution to the system?

• (0, 3)
• (1, -1)
• (-3, 1)
• (1, 3)

Multiple Choice

^{What is the solution?} 

• 1
• -2
• (1, 2)
• (1, -1)

How can we find the solution to this system of equations?

1. We can graph this by hand

or

2. We can graph this with desmos

Multiple Choice

What type of solution does this system have?

y = x + 2

y = 3x - 2

• One solution

• No solution

• Infinite solutions

Multiple Choice

What type of solution does this system have?

y = 2x + 2

y = 2x - 1

• One solution

• No solution

• Infinite solutions

Multiple Choice

What type of solution does this system have?

x - 2y = - 2

y = 1/2x + 1

• One solution

• No solution

• Infinite solutions

Fill in the Blanks

What is the solution to this system of equations?

y = x + 2

y = 3x - 2

Fill in the Blanks

What is the solution to this system of equations?

y = 2x + 4

y = x + 2

Multiple Choice

Solve the following system by graphing.  What is the solution?

• Infinite number of solutions
• (3, 3)
• (3, -3)
• (-3, 3)

Systems of Equations by Substitution

The solution (point of intersection) is (-2,-9)

Multiple Choice

What is the first step in solving a system by Substitution?

• Make sure both equations are in standard form.
• Make sure at least one equation is solved for one variable.
• Make sure both equations are in slope-intercept form.
• Make sure both equations can be solved.

Multiple Choice

Solve the following systems of equations using substitution:

x = 6

y = 2x - 3

• (6, 6)

• (6, 9)

• (9, 6)

• (9, 9)

Multiple Choice

Solve the following systems of equations using substitution:

5x - 2y = 3

y = 2x

• (6, 3)

• (1, 2)

• (3, 6)

• (2, 1)

Multiple Choice

Solve the system of equations using substitution

x=-6y

2x+12y=0

• (-6, 1)

• (0, 0)

• (1, -6)

• No solution

• Infinite solutions

Multiple Choice

Solve this system of equations. 
y = 2x + 1
y = 4x - 1

• (1,3)
• (-1,-3)
• (-1,3)
• (3,1)

Multiple Choice

Solve the system by substitution.
 
5x + 4y= −14
y =  −7x  −  15 

• (-2, -1)
• (1, -2)
• (-2, 1)
• (-1, -2)

Systems of Equation - Elimination

***VERY IMPORTANT SLIDE***

ADDITIVE INVERSES (opposites)

• WHEN YOU HAVE TWO OPPOSITE TERMS

• ONE IS POSITIVE AND THE OTHER IS NEGATIVE

• COMBINED (ADDED TOGETHER) EQUAL ZERO

• 3 + (-3) = 0

• 7y + (-7y) = 0

Multiple Choice

Which variable has opposite coefficients?

• X

• Y

Multiple Choice

Which variable can we ELIMINATE?

$$5x\ +\ 8y\ =\ 23$$   $$-5x\ +\ 4y\ =\ 5$$  

• X

• Y

Multiple Choice

Which variable can we ELIMINATE?

$$3x\ +\ -4y\ =\ -5$$   $$-10x\ +\ 4y\ =\ 12$$  

• X

• Y

Multiple Choice

When you combine (ADD) these 2 equations, what do you end up with? 

$$3x\ +\ -4y\ =\ -5$$   $$-10x\ +\ 4y\ =\ 12$$  

• $$-13x\ +\ 8y\ =\ 7$$  

• $$-7x\ =\ 7$$

• $$7x\ +\ 8y\ =\ 7$$  

WHEN SOLVING USING ELIMINATION

• Look for opposite coefficients

• IF THERE IS NONE, MULTIPLY EVERY TERM IN AN EQUATION

Multiple Choice

Solve using elimination. 
 4x + 8y = 20
-4x + 2y = -30

• (-7,1)
• (2,-5)
• (-2,5)
• (7,-1)

Multiple Choice

Solve the system by elimination.  

−8x − 10y = 20
−8x − 6y = −4 

• Infinite number of solutions  
• (6, 5) 
• (−6, −5) 
• (5, −6) 

Multiple Choice

-4x - 6y = 6
4x + 6y = -4

• no solution
• (2,0)
• (-4,0)
• (0,0)

Systems of Equations Word Problems

Multiple Choice

At a restaurant the cost for a breakfast taco and a small glass of milk is $2.10. The cost for 2 tacos and 3 small glasses of milk is $5.15. If a system of equations is written, which would be the correct representation of the 2 variables? 

• t: # of tacos
  m: # of glasses of milk
• t: Cost of each taco
  m: Cost of each glass of milk
• t: total cost
  m: # of food items
• t: Cost of each glass of milk
  m: Cost of each taco

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 Joe bought 10 cups of coffee and 5 doughnuts for his office at the cost of $16.50.  It turns out that the doughnuts were more popular than the coffee.  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 the coffee? 

• 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

Dennis mowed his next door neighbor’s lawn for a handful of dimes and nickels, 80 coins in all.  Upon completing the job he counted out the coins and it came to $6.60.  Which system of equations could be used to find the exact number of dimes and nickels? 

• d + n = 6.60
  .10d + .05n = 80
• d + n = 80
  d + n = 6.60
• d + n = 80
  .10d + .05n = 6.60
• d + n = 80
  .05d + .10n = 6.60

Fill in the Blanks

You're buying a new car.

A Toyota Prius costs $24,525 up front and will cost you $43.10 in gas each month.

A Honda Civic costs $21,250 up front and will cost you $78.13 in gas each month.

How many MONTHS will it take for the cost of both cars to be the same?

Round to three decimal places.

Fill in the Blanks

You're buying a new car.

A Toyota Prius costs $24,525 up front and will cost you $43.10 in gas each month.

A Honda Civic costs $21,250 up front and will cost you $78.13 in gas each month.

What is the COST when both cars cost the same?

Round to TWO decimal places.