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Algebra 1

2.2.3 Graphs of Systems of Equations

Algebra 12.2.3 Graphs of Systems of Equations

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Student Activity

Now that you have written equations from tables, it is time to write systems of equations from graphs.

Part 1: Writing a System of Equations from a Given Graph

For questions 1 – 8, use the given graph:

A graph with two lines: a solid orange line rising steeply from left to right and a dashed green line falling from left to right. Both lines intersect near the center of the coordinate grid.
1.

For the green dashed line what is the yy-value of the yy-intercept?

2.

For the green dashed line what is the slope?

3.

What is the equation of the green dashed line in slope-intercept form?

4.

For the orange solid line what is the yy-value of the yy-intercept?

5.

For the orange solid line what is the slope?

6.

What is the equation of the orange solid line in slope-intercept form?

7.

Write the two equations as a system of linear equations.

8.

What is the solution of the system of equations?

Part 2: Use a Graph to Solve a Given System

Now that you wrote a system of equations from a given graph, how can you work backward and graph a given system of equations by hand? Discuss this with a partner.

For questions 1 - 2, use this system of equations:

y=4x4y=4x4

y=2x+8y=2x+8

1.

On the same coordinate plane, graph y=4x4y=4x4 and y=2x+8y=2x+8 by hand.

2.

What is the solution to the system of equations?

For numbers 3 and 4, consider the system of equations:

y=34x+5y=34x+5

y=12x5y=12x5

3.

On the same coordinate plane, graph the system of equations by hand.

4.

What is the solution to the system of equations?

Video: Graphing a System of Equations

Watch the following video to learn more about writing systems of equations from graphs.

Are you ready for more?

Extending Your Thinking

Graph the following system of equations on a coordinate plane:

y=2x+5y=2x+5

4x2y=104x2y=10

Find the solution.

(Hint: For the second equation, solve for yy first.)

Self Check

Which of the following systems of equations is graphed:

  1. y = 3 x + 4

    y = 3 x + 8

  2. y = 4 x 4

    y = 8 x + 3

  3. y = 3 x + 4

    y = 3 x + 8

  4. y = 3 x + 4

    y = 3 x + 8

Additional Resources

Solving Systems by Graphing

The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.

Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.

Three graphs of linear equations: the first shows intersecting lines (one solution), the second parallel lines (no solution), and the third overlapping lines (infinitely many solutions), each labeled with a brief explanation.

Part 1:Writing a System of Equations from a Given Graph

How to write a system of linear equations from graphs.

Step 1 - Find the slope and yy-intercept of the first line.

Step 2 - Find the slope and yy-intercept of the second line.

Step 3 - Write the two equations for a system of equations.

Example 1

Write a system of equations from the graph.

A graph with a solid blue line sloping upward and a dashed red line with a gentler upward slope. The blue line crosses the y-axis at negative 3; the red line crosses at negative 3. Gridlines and labeled axes are shown.

Step 1 - Find the slope and yy-intercept of the redred dotteddotted line.

m=(y1y2)(x1x2)=(0(3))(60)=36=12m=(y1y2)(x1x2)=(0(3))(60)=36=12

yy-intercept = (0,3)(0,3)

Step 2 - Find the slope and yy-intercept of the blueblue solidsolid line.

m=(y1y2)(x1x2)=(3(3))(30)=63=21m=(y1y2)(x1x2)=(3(3))(30)=63=21

yy-intercept = (0,3)(0,3)

Step 3 - Write the two equations for a system of equations.

y=mx+by=mx+b

y=12x3y=12x3

y=2x3y=2x3

Part 2: Use a Graph to Solve a Given System

How to solve a system of linear equations by graphing.

Step 1 - Find the slope and yy-intercept of the first equation.

Step 2 - Find the slope and yy-intercept of the second equation.

Step 3 - Graph the two equations on the same rectangular coordinate system.

Step 4 - Estimate the solution to the system.

If the lines intersect, estimate the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system.

  • If the lines are parallel, the system has no solution.
  • If the lines are the same, the system has an infinite number of solutions.

Example 2

Solve the system by graphing:

y=2x+1y=2x+1

y=4x1y=4x1

Step 1 - Find the slope and /(y/)-intercept of the first equation.

In y=2x+1y=2x+1, m=2m=2, b=1b=1

Step 2 - Find the slope and yy-intercept of the second equation.

In y=4x1y=4x1, m=4m=4, b=1b=1

Step 3 - Graph the two equations on the same rectangular coordinate system.

A graph showing a solid orange line and a dashed teal line intersecting near point (1, 3), which is marked with a black dot and labeled. The x- and y-axes are visible with gridlines.

Step 4 - Estimate the solution to the system.

The point where the lines intersect is:

(1,3)(1,3)

Check to make sure the estimated point, (1,3)(1,3) solves the system by substituting the values for xx and yy into both equations.

Check the first equation: 𝑦=2𝑥+1𝑦=2𝑥+1

x=1x=1

y=3y=3

y=2x+1y=2x+1

(3)=2(1)+1(3)=2(1)+1

3=2+13=2+1

3=33=3

So, (1,3)(1,3) solves y=2x+1y=2x+1

Check the second equation: 𝑦=4𝑥1𝑦=4𝑥1

y=4x1y=4x1

(3)=4(1)1(3)=4(1)1

3=413=41

3=33=3

So, (1,3)(1,3) solves y=4x1y=4x1

Try it

Try It: Solving Systems by Graphing

1.

Write a system of equations from the graph.

A graph with a solid blue line and a dashed red line intersecting near the point (1,1) on a grid with labeled x- and y-axes. The blue line is steep and rising, the red line is descending.

2.

Solve the system by graphing:

y = 3 x + 3 y = 3 x + 3

y = x + 7 y = x + 7

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