Systems of Equations

​MGSE8.EE.8 Analyze and solve pairs of simultaneous linear equations

​Systems of Equations

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

Fill in the Blanks

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 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

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

• (4,2)

• (-2,1)

• No Solution

• Infinite Solutions

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 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.

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 Select

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

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

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

• (1,-4)

• (-1,-4)

• (-4,1)

• (-4,-1)

Multiple Choice

If solving by elimination, what variable would be the BEST to eliminate?

4x + 8y = 20

and −4x + 2y = −30

• X because they have opposite coefficients

• Y because they have opposite coefficients

Multiple Choice

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?

• Multiply the second equation by 2 and the result to the first equation.

• Multiply the second equation by -2 and the result to the first equation.

• Multiply the first equation by 7 and the second equation by -5 and add the resulting equations.

• Multiply the first equation by -7 and the second equation by -5 and add the resulting equations.

Fill in the Blanks

Multiple Choice

Solve for x and y:

3x + 2y = 16 and 7x + y = 19

• (-2,5)

• (-2,-5)

• (2,-5)

• (2,5)

Fill in the Blanks

Multiple Choice

What is the proper way to set up the equation to begin solving if using the substitution method?

y = 6x − 11

and −2x − 3y = −7

• -2(-11) -3y= -7

• -2(6x-11) -3y = -7

• -2x -3(11) = -7

• -2x - 3(6x-11)= -7

Fill in the Blanks

Fill in the Blanks

Fill in the Blanks

Multiple Choice

Determine if (4, 1) is a solution for the system of equations.
y = -x + 5
y = 2x - 7

• yes

• no

Multiple Choice

Solve the system:

5x + 3y = 7 and y = 4

• (4,1)

• (4,-1)

• (1,4)

• (-1,4)

Multiple Choice

Solve the system:

3x - y = 7

and 2x + y = 3

• (-1,2)

• (5,4)

• (4,5)

• (2,-1)

• (2,5)

​MGSE8.EE.8a Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Match

Match the following graphs to the systems of equations

 

 

  

   

• $$y=\frac{2}{3}x+1$$  

  $$y=\frac{2}{3}x-2$$  

• $$y=-x+7$$   $$y=2x+1$$  

• $$y=-2x+4$$   $$y=-2x+4$$   

Multiple Choice

Which of the following graphs would fit the system of equations?

$$y=2x-4$$  

$$y=-\frac{1}{2}x+1$$  

Multiple Choice

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

• Infinite number of solutions

• (3, 3)

• (3, -3)

• (-3, 3)

Graphing

Graph the solution to the system. The first equation has been plotted for you.

y = -3x-3

y= 4x +4

Graphing

Graph the solution to the system. The first equation has been plotted for you.

x = 1

y = 2x -4

Graphing

Graph the solution to the system. The first equation has been plotted for you.

x = 3

y= -1/3x + 3

Graphing

Graph the soluton to the system. The first line has been plotted for you.

y= -1/2x -1

y= 1/4 x -4

Graphing

Graph the solution to the system. The first equation has been plotted for you.

y= 4x + 3

y = -x -2

MGSE8.EE.8b Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations. Solve simple cases by inspection.

Multiple Choice

For which system of equations would you need to estimate the solution?

Multiple Choice

Sven determined that the x-coordinate is approximately 3.6 because the point is closer to 4 than 3 and seems to be a little more than halfway between them.

What is the approximate value for the y-coordinate?

• $$y\approx-1.1$$

• $$y\approx-1.4$$

• $$y\approx-1.8$$

• $$y\approx-1.9$$

Multiple Select

Sadira estimated the solution of a system of equations is approximately (–3.7, 2.2). What can be concluded of the graph that represents the system of this solution? Check all that apply.

• The x-coordinate of the intersection point is between 2 and 3.

• The y-coordinate of the intersection point is between 2 and 3.

• The y-coordinate of the intersection point is between -3 and -4.

• The x-coordinate is closer to –4 than –3.

• The y-coordinate is closer to –4 than –3.

Multiple Choice

Which graph shows a system of equations with an estimated solution of (–3.4, 1.2)?

Multiple Choice

Which graph represents the system of equations –3x + y = 3 and –x + 2y = –4?

Draw

Graph the system of equations AND state the solution:

y = 1/2 x + 4

y = -2x - 6

Draw

Graph the system of equations:

y = 3x + 2

y = -2x + 7

Circle what you think the solution would be!

Look at the next 2 questions. Determine the correct solution for each problem.

Use any method to solve the systems.

Multiple Select

Solve the systems with any method you choose: Check all that apply:

• x= -5/2

• y=-5/2

• No Solutions

• Infinitely Many Solutions

• x=0

Multiple Choice

Solve the system of equations

y = 2x + 1 and y = 4x - 1

• (1,3)

• (-1,-3)

• (-1,3)

• (1,-3)

​MGSE8.EE.8c Solve real-world and mathematical problems leading to two linear equations in two variables.

Multiple Choice

David is running a concession stand at a soccer game. He sells nachos (x) and sodas (y). Nachos cost $1.50 each and sodas cost $0.50 each. At the end of the game, David made a total of $78.50 and sold a total of 87 nachos and sodas combined. Which system of equations represents this situation?

• 1.5x+0.5y=78.5

  x+y=87

• 1.5x+0.5y=78.5

  1.5x+0.5y=87

• x+y=78.5

  1.5x+0.5y=87

• x+y=78.5

  x+y=87

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.  How long does it take her to do a haircut?
3x + 2y = 315
2x + 4y = 450

• 45 minutes

• 90 minutes 2x + 4y = 315

• 60 minutes 3x + 4y = 450

• 30 minutes

Multiple Choice

Gracie sells tickets for admission to your school play and collects a total of $104. Admission prices are $6 for adults and $4 for children. She sold 21 tickets total. How many of each type of ticket did Gracie sell?

• 10 adults, 11 children

• 12 adults, 9 children

• 11 adults, 10 children

• 9 adults, 12 children

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

Christian had brochures printed for a new business venture. Christian originally ordered 4 boxes of black-and-white brochures and 3 boxes of color brochures, which cost a total of $134. After those ran out, Christian spent $120 on 3 boxes of black-and-white brochures and 3 boxes of color brochures. Which system represents this situation?

• x+y=134 x+y=120

• 3x+3y=134 4x+3y=120

• 4x+3y=134 3x+3y=120

• 7xy=134 6xy=120

Multiple Choice

Amy sold 10 packages of sugar cookies and 2 packages of oatmeal cookies for $56. Adam sold 9 packages of sugar cookies and 3 packages of oatmeal cookies for $60. What is the price of 1 pack of sugar cookies and 1 pack of oatmeal cookies?

• Sugar Cookies $5  Oatmeal Cookies $3

• Sugar Cookies $4 Oatmeal Cookies $8

• Sugar Cookies $2 Oatmeal Cookies $6

• Sugar Cookies $3 Oatmeal Cookies $6