Systems of Equations - Graphical Method

In these lessons, we will learn how to solve systems of equations or simultaneous equations by graphing.

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Solving Equations
Solve Systems of Equations by Substitution
Solve Systems of Equations by Addition
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We have included a systems of equations calculator that can solve systems of equations graphically and algebraically. Use it to check your answers.

How to solve System of Equations by Graphing?

To solve systems of equations or simultaneous equations by the graphical method, we draw the graph for each of the equation and look for a point of intersection between the two graphs. The coordinates of the point of intersection would be the solution to the system of equations.

If the two graphs do not intersect - which means that they are parallel - then there is no solution.

Using the graphical method, find the solution of the systems of equations
y + x = 3
y = 4x - 2

Draw the two lines graphically and determine the point of intersection from the graph.
From the graph, the point of intersection is (1, 2)

Graphical Method - System of Equations

How to Solve Systems of Equations Graphically?

A system of equations is a set of two or more equations that are to be solved simultaneously.

A solution of a system of two equations in two variables is an ordered pair of numbers that makes both equations true. The numbers in the ordered pair correspond to the variables in alphabetical order.

What can happen when two lines are graphed on the same coordinate plane?

  1. The graphs intersect at one point. The system is consistent and has one solution. Since neither equation is a multiple of the other, they are independent.
  2. The graphs are parallel. The system is inconsistent because there is no solution. Since the equations are not equivalent, they are independent.
  3. Equations have the same graph. The system is consistent and has an infinite number of solutions. The equations are dependent since they are equivalent.


  1. Solve this system of equations by graphing:
    y = 3x + 1
    x - 2y = 3

  2. Solve this system of equations by graphing:
    y - x = 5
    2x - 2y = 10

  3. Solve this system of equations by graphing:
    y = 3x + 1
    x - 2y = 3

  4. Solve this system of equations by graphing:
    y = -x + 3
    2x - 2y = 10

How to solve systems of equations using the graphical method?

Systems of equations with one solution, no solutions (inconsistent system) and infinite solutions (dependent systems)


  1. Solve
    x + y = 1
    x - y = -5

  2. Solve
    y = 2x -4
    y = -1/2 x + 1

  3. Solve
    2x + 3y = 6
    y = -2/3 x - 2

Solving a Linear System of Equations by Graphing

The basic idea is to graph the two lines and look for any points of intersection.

5x - y = 6
2x + y = 8

Systems of Equations Calculator

This math tool will determine the intersection point of two lines or curves. Enter in the two equations and submit. The graphs of the two equations will be shown. Select step-by-step solution if you want to see the equations solved algebraically.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
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