Adding and Subtracting Systems of Equations and Using Their Graphs

2.4.3 Adding and Subtracting Systems of Equations and Using Their Graphs

2.4.3 • Adding and Subtracting Systems of Equations and Using Their Graphs

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

${\begin{matrix} 4 x + 3 y = 10 \\ - 4 x + 5 y = 6 \end{matrix}$

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

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Compare your answer:

Graph showing two intersecting lines: a red line with positive slope and a blue line with negative slope, on an x-y coordinate plane with axes labeled x and y.

Identify the $x$ coordinate of the solution.

Identify the $y$ coordinate of the solution.

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

Compare your answer:

Your answer may vary, but here are some samples.

Sample answer questions 1: $8 y = 16$
Sample answer questions 2: $y = 2$
Sample answer questions 3: $8 x - 2 y = 4$

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

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Compare your answer:

Graph of a system of three linear equations. The third graphed equation is the result of adding the first two equations together.

Make an observation about the graph.

Compare your answer:

Your answer may vary, but here is a sample.

All three lines intersect at $(1, 2)$ . If students graph $8 y = 16$ , the graph of the new equation is a horizontal line. If students graph $8 x - 2 y = 4$ , the graph of the new equation is a new line between the two original lines.

Use System B to answer questions 7 – 12.

System B

${\begin{matrix} 2 x + y = 4 \\ x - y = 11 \end{matrix}$

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

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Graph of a system of two linear equations. The orange line has a negative slope and the green line has a positive slope. The equations intersect in the fourth quadrant.

Identify the $x$ coordinate of the solution.

Identify the $y$ coordinate of the solution.

-6.

10.

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

Compare your answer:

Your answer may vary, but here are some samples.

Sample answer questions 1: $3 x = 15$
Sample answer questions 2: $x = 5$

11.

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

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Compare your answer:

12.

Make an observation about the graph.

Compare your answer:

Your answer may vary, but here is a sample. All three lines intersect at $(5, - 6)$ . If students graph $3 x = 15$ , the graph of the new equation is a vertical line.

Use System C to answer questions 13–18.

System C

${\begin{matrix} 8 x + 11 y = 37 \\ 8 x + y = 7 \end{matrix}$

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.

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Compare your answer:

Graph of a system of two linear equations. The orange line has a negative slope and the green line has a steeper, negative slope. The equations intersect in the first quadrant.

14.

Identify the $x$ coordinate of the solution.

Compare your answer:

$\frac{1}{2}$

15.

Identify the $y$ coordinate of the solution.

16.

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

Compare your answer:

Your answer may vary, but here are some samples.

Sample answer questions 1: $10 y = 30$
Sample answer questions 2: $- 10 y = - 30$
Sample answer questions 3: $y = 3$
Sample answer questions 4: (If the equations are added) $16 x + 12 y = 44$

17.

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

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Compare your answer:

18.

Make an observation about the graph.

Compare your answer:

Your answer may vary, but here is a sample. All three lines intersect at $(\frac{1}{2}, 3)$ . If students graph $10 y = 30$ , the graph of the new equation is a horizontal line.

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

${\begin{matrix} 2 x + y = 4 \\ x - y = 11 \end{matrix}$

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

Compare your answer:

Your answer may vary, but here is a sample. $(2 x + y) (x - y) = 44$

Does your original solution still work in this new equation?

Compare your answer:

Yes

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

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Compare your answer:

Graph of a system of four equations. The blue line has a positive slope, the black line has a negative slope, the red line is vertical, and the green curve represents a non-linear equation. All of the equations intersect at the point (5, 6).

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

Compare your answer:

This approach is not helpful because the new equation is not a line and therefore not so easy to work with.

Self Check

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

$\begin{array}{rcl}&&\left\{\begin{array}{l}2x\;–\;4y\;=\;–26\\-3x\;+\;4y\;=\;29\end{array}\right.\end{array}$

$(-13, 0)$
$(0, 6.25)$
$(-3, 5)$
$(-8, 2.5)$

Additional Resources

Adding Systems of Equations and Using Their Graphs

Look at the system of equations shown here:

$\begin{matrix} {\begin{matrix} - 3 x + 4 y = 7 \\ 3 x + 2 y = 17 \end{matrix} \end{matrix}$

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 $(3, 4)$ .

Add the two equations together. Simplify the resulting equation.

$\begin{matrix} - 3 x + 4 y = 7 \\ 3 x + 2 y = 17 \\ 6 y = 24 \\ y = 4 \end{matrix}$

Graph the resulting equation on the same coordinate plane.

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

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

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 $x = 3$ , $y = 4$ .

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.

${\begin{matrix} - 2 x - 3 y = 7 \\ 2 x + y = 3 \end{matrix}$

Add the equations to find the resulting equation.

Compare your answer:

$\underset{―}{- 2 x - 3 y = 7 + 2 x + y = 3} - 2 y = 10 y = - 5$

$\begin{matrix} 2 x + y & = & 3 \\ 2 x + (- 5) & = & 3 \\ 2 x & = & 8 \\ x & = & 4 \end{matrix}$

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.

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Compare your answer:

Graph of three lines that intersect at the point (4,-5).

Identify the $x$ coordinate of the solution.

Identify the $y$ coordinate of the solution.

-5.