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:
For the green dashed line what is the -value of the -intercept?
9
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:
For the orange solid line what is the -value of the -intercept?
5.
For the orange solid line what is the slope?
3.
What is the equation of the orange solid line in slope-intercept form?
Compare your answer:
Write the two equations as a system of linear equations.
Compare your answer:
What is the solution of the system of equations?
Compare your answer:
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:
On the same coordinate plane, graph and by hand.
Compare your answer:
What is the solution to the system of equations?
Compare your answer:
For numbers 3 and 4, consider the system of equations:
On the same coordinate plane, graph the system of equations by hand.
Compare your answer:
What is the solution to the system of equations?
Compare your answer:
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:
Find the solution.
(Hint: For the second equation, solve for first.)
Compare your answer:
Infinitely many solutions. When the second equation is solved for , it becomes then . Both equations are the same. So, on the graph, one line is on top of the other and all solutions are possible.
Self Check
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.
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 -intercept of the first line.
Step 2 - Find the slope and -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.
Step 1 - Find the slope and -intercept of the line.
-intercept =
Step 2 - Find the slope and -intercept of the line.
-intercept =
Step 3 - Write the two equations for a system of equations.
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 -intercept of the first equation.
Step 2 - Find the slope and -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:
Step 1 - Find the slope and /(y/)-intercept of the first equation.
In , ,
Step 2 - Find the slope and -intercept of the second equation.
In , ,
Step 3 - Graph the two equations on the same rectangular coordinate system.
Step 4 - Estimate the solution to the system.
The point where the lines intersect is:
Check to make sure the estimated point, solves the system by substituting the values for and into both equations.
Check the first equation:
So, solves
Check the second equation:
So, solves
Try it
Try It: Solving Systems by Graphing
Write a system of equations from the graph.
Compare your answer:
Here is how to solve the system by graphing:
Step 1 - Find the slope and -intercept of the line.
Step 2 - Find the slope and -intercept of the line.
Step 3 - Write the two equations for a system of equation.
Solve the system by graphing:
Compare your answer:
Here is how to solve the system by graphing:
Step 1 - Find the slope and -intercept of the first equation.
In , ,
Step 2 - Find the slope and -intercept of the second equation.
In , ,
Step 3 - Graph the two equations on the same rectangular coordinate system.
Step 4 - Estimate the solution to the system.