Solving systems of equations by graphing worksheet answer key

The worksheets on this page have four coordinate planes and systems equations in slope intercept form that students graph to solve, and includes an answer key showing the correct graph.

Graphing Systems of Equations

What are Systems of Equations?

Two or more linear equations that are related are called a system of equations. Graphing systems of equations involves graphing each individual linear equation in the system. The places where the lines intersect represent solutions where two or more of the linear equations share a common solution, and that point is regarded as the solution to the entire system.

You can solve a system of equations by graphing the lines and seeing where they intersect. This is called “solving by graphing” and is a valid approach for linear equations with relatively simple slope and y-intercept values. The graphing systems of equations worksheets on this page fit this criteria and they are good practice for building a visual intuition of the solution process.

In practice, it the linear equations in a system are more complicated and attempting to determine an accurate solution by graphing is limited by how easily readable on each axis. Generally, these solutions should be regarded as approximate except in cases where the slopes and intercepts in the equations are small integers and the solution is obviously correct for both equations. Even then, manually checking the solution algebraically is still a sound verification.

In practice, it is more common to solve systems of equations by substition. Typically your system of equations will include two equations in slope intercept form, with both equations as a y value calculated in terms of x. Solving by substition involves combining the two equations into a single function that gives either a x or y coordinate. You can do this easily by replacing the y side of one of the equations with the mx+b expression from the other (so, “mx+b = mx+b”) and the solving for x. This will give you an x value that can then be substituted into either of the original equations to calculate a corresponding y coordinate. The resulting x and y values make up the coordinate of a solution to both equations, and therefore a solution to the combined system of equations.

Solving Systems of Equations by Graphing

You can solve systems of equations by graphing using the following steps:

1. For each, equation graph the line. Click here if you need help or practicing graphing linear equations .


2. If the lines do not intersect (they are parallel), then the system of equations has no solution.
3. If the lines do intersect, find the coordinates of the point where the lines from each equation intersect.
4. The intersection point is a shared solution to both equations, and therefore a solution to the entire system of equations.

While this approach is arguably more intuitive compared to solving by substition, it can also be less precise. Again, it’s recommeded that students check solutions obtain from solving a system equations by graphing by plugging the x coordinate from the solution into each equation and verifying that the calculated y value from each equation in the system winds up being the same.

If you are graphing linear equations, the worksheets on this page provide great practice resources for middle school algebra students. You can also print a blank coordinate plane to graph other equations, or try working with the slope calculator to see how different points are used to calculate slope and find equations in slope intercept form.

Problem 1 : 

Solve the given system of equations by graphing. 

x + y - 4  =  0 

3x - y  =  0

Problem 2 : 

Solve the given system of equations by graphing. 

3x - y - 3  =  0 

x - y - 3  =  0

Problem 3 : 

Lily and her friends visit the concession stand at a football game. $2 is charged for a sandwich and $1 is charged for a lemonade by the stand. The friends buy a total of 8 items for $11. Say how many sandwiches and how many lemonades they bought.

Answers

1. Answer :

x + y - 4  =  0 

3x - y  =  0

Step 1 :

Let us re-write the given equations in slope-intercept form  (y = mx + b). 

y  =  - x + 4

(slope is -1 and y-intercept is 4) 

y  =  3x 

(slope is 3 and y-intercept is 0) 

Based on slope and y-intercept, we can graph the given equations. 

Step 2 :

Find the point of intersection of the two lines. It appears to be (1, 3). Substitute to check if it is a solution of both equations.

x + y - 4  =  0 

1 + 3 - 4  =  0  ?

4 - 4  =  0  ?

0  =  0  True

3x - y  =  0 

3(1) - 3  =  0  ?

3 - 3  =  0  ?

0  =  0  True

Because the point (1, 3) satisfies both the equations, the solution of the system is (1, 3).  

2. Answer :

3x - y - 3  =  0 

x - y - 3  =  0

Step 1 :

Let us re-write the given equations in slope-intercept form. 

y  =  3x - 3

(slope is 3 and y-intercept is -3) 

y  =  x - 3 

(slope is 1 and y-intercept is -3) 

Based on slope and y-intercept, we can graph the given equations. 

Step 2 :

Find the point of intersection of the two lines. It appears to be (0, -3). Substitute to check if it is a solution of both equations.

3x - y - 3  =  0 

3(0) - (-3) - 3  =  0  ?

0 + 3 - 3  =  0  ?

0  =  0  True

x - y - 3  =  0

0 - (-3) - 3  =  0  ?

3 - 3  =  0  ?

0  =  0  True

Because the point (0, -3) satisfies both the equations, the solution of the system is (0, -3).  

3. Answer :

Step 1 :

Let "x" be the no. of sandwiches and "y" be the no. of lemonades. 

No. of sandwiches + No. of lemonades  =  Total items

x + y  =  8 -------- (1)

Step 2 : 

Write an equation which represents the money spent on the items.

Cost of "x" no. of sandwiches  =  2x

Cost of "y" no. of sandwiches  =  1y

Total cost  =  $11

Then, we have 

2x + 1y  =  11 -------- (2)

Step 3 :

Write the equations (1) and (2) in slope-intercept form. Then graph.

(1) ----- > x + y  =  8 ----- > y  =  -x + 8 

(2) ---- > 2x + 1y  =  11 --- > y  =  -2x + 11 

Graph the equations y = -x + 8 and y = -2x + 11.

Step 4 :

Look at the graph and identify the solution of the system of equations. Check your answer by substituting the ordered pair into both equations.

The point of intersection of the lines is (3, 5). 

Let us check whether the ordered pair (3, 5) satisfies both the equations. 

 y  =  -x + 8 

5  =  -3 + 8

5  =  5

y  =  -2x + 11

5  =  -2(3) + 11

5  =  -6 + 11

5  =  5

Since the ordered pair (3, 5) satisfies both the equations, the solution of the system is (3, 5). 

Step 5 : 

Interpret the solution in the original context.

Lily and her friends bought 3 sandwiches and 5 lemonades.

Kindly mail your feedback to 

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com