How to Solve Simultaneous Equations Graphically

Reviewed by Joseph Meyer

Last Updated: April 13, 2024 Approved

There are a few ways to solve simultaneous equations;^{[1]
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Research source} one of them is to plot the lines of the 2 equations, thus solving the equation. This article will teach you how to solve simultaneous equations graphically!

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1
Write your 2 equations clearly. Write your 2 equations in an organised, clear way. This will make it easier later.^{[2]
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- In this guide, we are going to use equations 3x+4y=22 and x+2y=2 as an example.
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In the first equation, let x be 0. Replace x with 0, and solve the equation. This should give you the value of y.
- Using the example before: 3*0+4y=22, which becomes 4y=22, and then y=5½
- You now have your first coordinate, (0 ; 5½)
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3
Let y be 0. In the first equation, let y be 0 and solve the equation. This will give you the value of x.
- Using the example: 3x+4*0=22 which becomes 3x=22 which gives us the answer of x=7⅓
- You now have a set of coordinates for your first line, (0 ; 5½) and (7⅓ ; 0)
Draw the line of your first equation on it.^{[3]
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EXPERT TIP

Joseph Meyer
Math Teacher
Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.

Joseph Meyer
Math Teacher

Develop strong graphing skills. Drawing graphs by hand will help you develop foundational graphing skills, especially in understanding scales and axes. This will build a strong base for you to use helpful online tools to visualize complex relationships, perform calculations, and prepare for standardized tests.
5
In the second equation, let x be 0. Then solve the equation, giving you the value of y.
- Using the example, it's 0+2y=2, which becomes 2y=2 and y=1
- First coordinate is now (0 ; 1)
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Let y be 0. Solve the equation and get the value of x.
- Example: x+2*0=2, which becomes x=2.
- Second coordinate is now (2 ; 0)
If needed, increase the length of the 2 lines until they meet.^{[4]
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Look at the point in which the 2 lines meet. Get it's coordinates (x ; y). The x coordinate will be the value of x in the equation, and the y coordinate will be the value of y!^{[5]
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- If you've been using the example, the lines should meet at point (18 ; -8). x will be 18, and y will be -8!

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What will be the answer if lines do not intersect?

Donagan

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If the graphed lines do not intersect, the system has no solution. That means there is no set of values for the variables that will work for both equations simultaneously.

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How do I guess the number, like x=0 or x=1, etc.?

Donagan

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You don't guess any numbers. You simply use a low number for x (like 0, 1, 2, -1, or -2, for example) so that it's easy to calculate the corresponding value of y.

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How do I solve y=2x+1, y=x-5?

Donagan

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Plot both equations on the same graph. The x and y values of the point where the two lines intersect are the solution of the equations.

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If you are not sure about the answer you get, check it! Do this by replacing x and y in the equation with the values you obtained.

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Always be precise when using this method: a slight imprecision could get you the wrong answer!

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