Graphs of Systems of Equations

2.2.3 Graphs of Systems of Equations

2.2.3 • Graphs of Systems of Equations

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.

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

For the green dashed line what is the slope?

-1

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

Compare your answer:

$y = - x + 9$

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

For the orange solid line what is the slope?

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

Compare your answer:

$y = 3 x + 5$

Write the two equations as a system of linear equations.

Compare your answer:

$y = - x + 9$

$y = 3 x + 5$

What is the solution of the system of equations?

Compare your answer:

$(1, 8)$

Are you ready for more?

Extending Your Thinking

Graph the following system of equations on a coordinate plane:

$y = 2 x + 5$

$4 x - 2 y = 10$

Find the solution.

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

Compare your answer:

Infinitely many solutions. When the second equation is solved for $y$ , it becomes $- 2 y = - 4 x = 10$ then $y = 2 x + 5$ . Both equations are the same. So, on the graph, one line is on top of the other and all solutions are possible.

Video: Graphing a System of Equations

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

Access multimedia content

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:

$y = 4 x - 4$

$y = - 2 x + 8$

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

Compare your answer:

A graph with two lines: a solid orange line with positive slope and a dashed green line with negative slope, both crossing the y-axis near the origin. The grid shows labeled x and y axes.

What is the solution to the system of equations?

Compare your answer:

$(2, 4)$

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

$y = - \frac{3}{4} x + 5$

$y = \frac{1}{2} x - 5$

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

Compare your answer:

A graph with a solid orange line increasing and a dashed blue line decreasing. The x- and y-axes are labeled, with a visible grid.

What is the solution to the system of equations?

Compare your answer:

$(8, - 1)$

Self Check

Which of the following systems of equations is graphed:

$y=3x+4$

$y=3x+8$
$y=4x-4$

$y=8x+3$
$y=3x+4$

$y=-3x+8$
$y=-3x+4$

$y=3x+8$

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 $y$ -intercept of the first line.

Step 2 - Find the slope and $y$ -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 $y$ -intercept of the $r e d$ $d o t t e d$ line.

$m = \frac{(y_{1} - y_{2})}{(x_{1} - x_{2})} = \frac{(0 - (- 3))}{(6 - 0)} = \frac{3}{6} = \frac{1}{2}$

$y$ -intercept = $(0, - 3)$

Step 2 - Find the slope and $y$ -intercept of the $b l u e$ $s o l i d$ line.

$m = \frac{(y_{1} - y_{2})}{(x_{1} - x_{2})} = \frac{(3 - (- 3))}{(3 - 0)} = \frac{6}{3} = \frac{2}{1}$

$y$ -intercept = $(0, - 3)$

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

$y = m x + b$

$y = \frac{1}{2} x - 3$

$y = 2 x - 3$

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 $y$ -intercept of the first equation.

Step 2 - Find the slope and $y$ -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:

$y = 2 x + 1$

$y = 4 x - 1$

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

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

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

In $y = 4 x - 1$ , $m = 4$ , $b = - 1$

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:

$(1, 3)$

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

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

$x = 1$

$y = 3$

$y = 2 x + 1$

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

$3 = 2 + 1$

$3 = 3$

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

Check the second equation: $𝑦 = 4 𝑥 - 1$

$y = 4 x - 1$

$(3) = 4 (1) - 1$

$3 = 4 - 1$

$3 = 3$

So, $(1, 3)$ solves $y = 4 x - 1$

Try it

Solving Systems by Graphing

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.

Compare your answer:

Here is how to solve the system by graphing:

Step 1 - Find the slope and $y$ -intercept of the $r e d$ $d o t t e d$ line.

$m = \frac{(y_{1} - y_{2})}{(x_{1} - x_{2})} = \frac{(0 - (1))}{(2 - 0)} = \frac{- 1}{2} = \frac{- 1}{2}$

Step 2 - Find the slope and $y$ -intercept of the $b l u e$ $s o l i d$ line.

$m = \frac{(y_{1} - y_{2})}{(x_{1} - x_{2})} = \frac{(1 - (- 3))}{(1 - 0)} = \frac{4}{1}$

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

$y = \frac{- 1}{2} x + 1$

$y = 4 x - 3$

Solve the system by graphing:

$y = 3 x + 3$

$y = - x + 7$

Compare your answer:

Here is how to solve the system by graphing:

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

In $y = 3 x + 3$ , $m = 3$ , $b = 3$

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

In $y = - x + 7$ , $m = - 1$ , $b = 7$

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

A graph with two intersecting lines: one orange and one teal dashed. They cross at the point (1, 6), which is labeled and marked with a black dot. The axes are labeled with gridlines and numbers.

Step 4 - Estimate the solution to the system.

$(1, 6)$