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Algebra 1

2.4.3 Adding and Subtracting Systems of Equations and Using Their Graphs

Algebra 12.4.3 Adding and Subtracting Systems of Equations and Using Their Graphs

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Activity

Access the Desmos guide PDF for tips on solving problems with the Desmos graphing calculator.

Below are three systems of equations you saw earlier.

Use System A to answer questions 1 – 6.

System A

{4x+3y=104x+5y=6{4x+3y=104x+5y=6

1.

Use the graphing tool or technology outside the course. Graph the equations for System A that represent this scenario using the Desmos tool below.

2.

Identify the xx coordinate of the solution.

3.

Identify the yy coordinate of the solution.

4.

Find the sum or difference of the two original equations that would enable the system to be solved.

5.

Use the graphing tool or technology outside the course. Graph your third equation and the System A equations using the Desmos tool below.

6.

Make an observation about the graph.

Use System B to answer questions 7 – 12.

System B

{2x+y=4xy=11{2x+y=4xy=11

7.

Use the graphing tool or technology outside the course. Graph the equations for System B that represent this scenario using the Desmos tool below.

8.

Identify the xx coordinate of the solution.

9.

Identify the yy coordinate of the solution.

10.

Find the sum or difference of the two original equations that would enable the system to be solved.

11.

Use the graphing tool or technology outside the course. Graph your third equation and the System B equations using the Desmos tool below.

12.

Make an observation about the graph.

Use System C to answer questions 13–18.

System C

{8x+11y=378x+y=7{8x+11y=378x+y=7

13.

Use the graphing tool or technology outside the course. Graph the equations for System C that represent this scenario using the Desmos tool below.

14.

Identify the xx coordinate of the solution.

15.

Identify the yy coordinate of the solution.

16.

Find the sum or difference of the two original equations that would enable the system to be solved.

17.

Use the graphing tool or technology outside the course. Graph your third equation and the System C equations using the Desmos tool below.

18.

Make an observation about the graph.

Are you ready for more?

Extending Your Thinking

Mai wonders what would happen if we multiply equations. That is, we multiply the expressions on the left side of the two equations and set them equal to the expressions on the right side of the two equations.

Use System B to answer questions 1 – 4.

System B

{2x+y=4xy=11{2x+y=4xy=11

1.

Write out an equation that you would get if you multiply the two equations in this manner.

2.

Does your original solution still work in this new equation?

3.

Use the graphing tool or technology outside the course. Graph your third equation and the System B equations using the Desmos tool below.

4.

Explain if the approach of graphing the new equation is or is not particularly helpful.

Self Check

Solve the system of equations using elimination. Check the solution by graphing.

{ 2 x 4 y = 26 3 x + 4 y = 29

  1. ( 13 , 0 )
  2. ( 0 , 6.25 )
  3. ( 3 , 5 )
  4. ( 8 , 2.5 )

Additional Resources

Adding Systems of Equations and Using Their Graphs

Look at the system of equations shown here:

{3x+4y=73x+2y=17{3x+4y=73x+2y=17

Let’s graph each equation on the coordinate plane. Identify the point of intersection.

Graph of two lines intersecting at the point (3, 4).

The lines intersect at (3,4)(3,4).

Add the two equations together. Simplify the resulting equation.

3x+4y=73x+2y=176y=24y=43x+4y=73x+2y=176y=24y=4

Graph the resulting equation on the same coordinate plane.

Graph of three lines intersecting at the point (3,4).

All three lines intersect at (3,4)(3,4).

You verified that by adding the two equations in the system, you created a third equation that is also true.

The solution to the system is x=3x=3, y=4y=4.

Try it

Try It: Adding Systems of Equations and Using Their Graphs

Add the equations to find the resulting equation. Then graph all three equations to find the ordered pair where they intersect.

Use this system of equations to answer questions 1 – 4.

{2x3y=72x+y=3{2x3y=72x+y=3

1.

Add the equations to find the resulting equation.

2.

Use the graphing tool or technology outside the course. Graph your third equation and the system of equations for this scenario using the Desmos tool below.

3.

Identify the xx coordinate of the solution.

4.

Identify the yy coordinate of the solution.

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