Lines of Intersection

Systems of Equations

What answer choices are possible when you draw 2 lines on the same graph?

When 2 lines are drawn on the same graph, there are 3 possible outcomes.

• The lines intersect

• The lines don't intersect

• The lines are on top of each other

Multiple Choice

Here is an example of 2 lines being graphed. What answer choice matches the situation shown?

• The liness intersect

• The lines do not intersect

• The lines are on top of each others

For lines not to intersect at all...

• The lines must be parallel

• If the lines are parallel, we can write 'NO SOLUTION'

• Parallel lines have the SAME SLOPE but DIFFERENT y-intercepts

Multiple Choice

Here is an example of 2 lines being graphed. What answer choice matches the situation shown?

• The liness intersect

• The lines do not intersect

• The lines are on top of each others

For lines to be right on top of one another...

• You should only see one line even though there are 2 equations.

• When this situation occurs, you can write "INFINITE SOLUTIONS"

• The 2 equations have to be mathematically the equal to one another for the lines to be on top of one another.

Multiple Choice

Here is an example of 2 lines being graphed. What answer choice matches the situation shown?

• The liness intersect

• The lines do not intersect

• The lines are on top of each others

For lines to intersect once...

• The place where the line intersects is at an ordered pair

• Write the orderd pair in the form of (x,y).

• The slopes of the two lines must be different.

Look at the next 6 questions. Determine the correct solution for each problem.

Choose the correct ordered pair as (x,y) if the lines cross once.

Choose NO SOLUTION if the lines are parallel. Choose INFINITE SOLUTIONS if the lines are on top of each other.

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (4,2)

• (-2,1)

• No Solution

• Infinite Solutions

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (4,2)

• (-2,1)

• No Solution

• Infinite Solutions

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (4,2)

• (-2,1)

• No Solution

• Infinite Solutions

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (1,-4)

• (-1,-4)

• (-4,1)

• (-4,-1)

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (1,-1)

• (-1,-1)

• (1,1)

• (-1,1)

Multiple Choice

What is the point of intersection for the problem at the left, if any?

• (-3,0)

• (0,3)

• (3,0)

• (0,-3)

Multiple Select

Choose the graph or graphs that have one solution. Check the box or boxes that apply.

Multiple Select

Choose the graph or graphs that have INFINITE SOLUTIONS. Check the box or boxes that apply.

Fill in the Blanks

Multiple Choice

What is the solution to the systems of equations shown in the graph?

• 1 Solution x=-1/2

• 1 Solution y=-1/2

• Many Solutions

• No Solution

You must watch the video on the next slide...

Graphing Systems Word Problems

• Read the text (then re-read the text)

• Write a Let Statement

• Set up two (2) equations

• Graph both equations on the same coordinate plane. USE DESMOS!!!!

• Look for the intersection point

• Answer the question

Multiple Choice

What is the solution to this system of equations?

• (1,4)

• (4,1)

• (3,5)

• (-1, 4)

Multiple Choice

Match the system of linear equations to the corresponding graph.

$$y=2x-5$$   $$y=-\frac{3}{2}x+2$$  

Multiple Choice

What is the solution to the system of linear equations?

• (0, 2)

• (-2, -2)

• (0, 1)

• (2, 0)

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

Multiple Choice

Last season two running backs on the Steelers football team rushed a combined total of 1550 yards. One rushed 4 times as many yards as the other. Let x and y represent the number of yards each individual player rushed. Which system of equations could be used?

• x + y = 1550

  y = 4x

• x + y = 1550

  y = x + 4

• y - x = 1550

  y = 4x

• y = 1550 + x

  y = x + 4

Multiple Choice

The school that Stefan goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Which equations represents the system that could be used?

• 1s + 3c = 38

  2s + 3c = 52

• 3s + 1c = 38

  3s + 2c = 52

• s + c = 38

  s + c = 52

• 3s + 3c = 38

  1s + 2c = 52

Multiple Choice

Josh is thinking of two numbers.  Their sum is -10 and their difference is -2.  Which system of equations represents the situation?

• x - y = -10 x + y = -2

• x = -2  y = 5

• x + y = -2 x - y = -10

• x + y = -10 x - y = -2

Multiple Choice

Nancy went to the grocery story.  On Monday she purchased 4 apples and 6 bananas for a total of $13.  On Wednesday she purchased 3 apples and 7 bananas for a total of $13.50.  Which system of equations represents the situation?

• 4x + 6y = 3 13.5x - 13y = 6

• x + y = 4 x - y = 6

• 4x + 6y = 13 3x + 7y = 13.5

• 4x - 6y = 13 3x - 7y = 13.5

Multiple Choice

The talent show committee sold a total of 530 tickets in advance. Student tickets cost $3 each and the adult tickets cost $4 each. If the total receipts were $1740, which system could be used to find how many of each type of ticket were sold?

• S + A = 530 3S + 4A = 1740

• S + A = 530 4S + 3A = 1740

• S + A = 1740 3S + 4A = 530

• S + A = 1740 4S + 3A = 530

Multiple Choice

Homer sells tickets for admission to your school play and collects a total of $104. Admission prices are $6 for adults and $4 for children. He sold 21 tickets total.

To solve, I graphed:

x + y = 21

6x + 4y = 104

I got an intersection point at (10,11). What does that mean in context of the problem?

• There are 10 adults and 11 children.

• There are 11 adults and 10 children.

• The adults are $10 and the children are $11.

• There are 104 adults and 21 children.

Multiple Choice

Flagship Cinemas sells movie tickets that are $4 for matinees and $7 for regular. One night, the theater sells 578 tickets and collects $3365 in total ticket sales.
Which system best represents the situation?

• 4x + 7y = 578 x + y = 3365

• 4x + 7y = 3365 x + y = 578

• 4y + 7x = 578 x + y = 3365

• 4y + 7x = 3365 x + y = 578

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

Caitlin won a bag full of money! She has 49 bills in all. She counts $1430. There are twenty dollar bills and fifty dollar bills. How many of each bill does Caitlin have?  Which system best represents the situation? 

• x + y = 1430 20x + 50y = 49

• x + y = 49 10x + 5y = 1430 

• x + y = 49 20x + 50y = 1430

• x + y = 49 x + y = 1430

Multiple Choice

Trevor and Kylie are selling fruit for a school fundraiser. Customers can buy small boxes of oranges (x) and large boxes of oranges (y). Trevor sold 3 boxes of small oranges and 14 boxes of large oranges for a total of $203. Kylie sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220.

• $$14x\ +\ 3y\ =\ 203$$ $$11x\ +\ 11y\ =\ 220$$

• $$3x\ +\ 14y\ =\ 203$$ $$11x\ +\ 11y\ =\ 220$$

• $$3x\ +\ 14y\ =\ 220$$ $$11x\ +\ 11y\ =\ 203$$

• $$14x\ +\ 3y\ =\ 220$$ $$11x\ +\ 11y\ =\ 203$$

Multiple Choice

Stacy went to the store to buy vegetables. The cost of 5 squash (x) and 2 zucchini (y) was $1.32. Three squash and 1 zucchini cost $0.75.

• $$5x\ +\ 2y\ =\ 1.32$$ $$3x\ +\ y\ =\ .75$$

• $$2x\ +\ 5y\ =\ 1.32$$ $$x\ +\ 3y\ =\ .75$$

• $$5x\ +\ 2y\ =\ .75$$ $$3x\ +\ y\ =\ 1.32$$

• $$2x\ +\ 5y\ =\ 1.32$$ $$x\ +\ y\ =\ .75$$

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

At a college bookstore, Carla purchased a math textbook and a novel that cost a total of $54, not including tax. If the price of the math textbook, t, is $8 more than 3 times the price of the novel, n, which system of linear equations could be used to determine the price of each book?

• t + n = 54 t = 3n + 8

• t + n = 54 n = 3t + 8

• t + n = 54 t = 3n - 8

• t + n = 8 t = 3n + 54

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.  Which system of equations represents the situation?

• 3x + 2y = 315 2x + 4y = 450

• 3x + 2y = 450 2x + 4y = 315

• 2x + 2y = 315 3x + 4y = 450

Multiple Choice

The cost of 5 squash and 2 zucchini is $1.32. Three squash and 1 zucchini cost $0.75. Write a system of equations.

• 5q + 2z = 1.32 1z = 0.75

• 5q + 2z = 1.32 3q + 1z = 0.75

• q + z = 1.32 q + z = 0.75

• 5q + 2z = 0.75 3q + 1z = 1.32

Multiple Choice

Your family goes to a restaurant for dinner. There are 6 people in your family. Some order the chicken dinner for $14.80, and some order steak for $17. If the total bill was $91, which system best represents the situation? 

• x + y = 6 14.80x + 17y = 91

• x + y = 91 14.80x + 17y = 6

• x + y = 6 14.80y + 17x = 91

• x + y = 91 14.80x+ 17y = 6