How to solve a system of equations containing two variables

For “system of equations” you will have to solve two or more equations at the same time. When these equations have two different variables, such as x and y or a and b, you may not know how to solve them at first. Luckily, all you need is basic algebra skills to solve a system of equations (sometimes a little knowledge of fractions is required). If you are in the habit of learning visually, or if your teacher requires it, you should learn how to graph equations. Graphing can be a useful way to “see what’s going on” or to check results, but this is slower than other solutions and doesn’t work for every system of equations.

Table of Contents

Use the alternative method

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Convert the variables in the equation. This “alternative” method begins by “solving for x” (or any other variable) of either equation. For example, say you have the equations 4x + 2y = 8 and 5x + 3y = 9 . Start with the first equation. Transform the equation by subtracting 2y from both sides to get: 4x = 8 – 2y .

This method often requires the use of fractions in the solution. You can use the elimination method below if you don’t like using fractions.

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Divide both sides of the equation to “solve x. “ After you’ve converted the term x (or whatever variable you’re using) to one side of the equation, divide both sides of the equation to leave only one. me x. For example:

4x = 8 – 2y
(4x)/4 = (8/4) – (2y/4)
x = 2 – y

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Substitute this result into the remaining equation. You have to substitute the other equation, not the one you used. In that equation, you substitute the variable you found so that the equation has only one variable. For example:

You know that x = 2 – ½y .
The second equation that hasn’t been transformed is 5x + 3y = 9 .
In the second equation, replace x with “2 – ½y”: 5(2 – ½y) + 3y = 9 .

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Solve for the remaining variable. You now have an equation containing only one variable. Use basic algebra skills to solve for that variable. If the variables cancel, go to the last step . Otherwise, you get the result for one of the system’s variables:

5(2 – y) + 3y = 9
10 – (5/2)y + 3y = 9
10 – (5/2)y + (6/2)y = 9 (If you don’t understand this step, learn how to add fractions. This is common knowledge in the substitution method, but not always. .)
10 + y = 9
y = -1
y = -2

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Use this answer to find the remaining variable. Don’t be confused when you’ve only solved half of the problem. You will have to substitute the result you just found in one of the original equations, so you can find the other variable:

You know that y = -2
One of the two original equations is 4x + 2y = 8 . (You can use either equation at this step.)
Substitute -2 for y: 4x + 2(-2) = 8 .
4x – 4 = 8
4x = 12
x = 3

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When two variables cancel, what to do? When you substitute x=3y+2 or a similar result into the other equation, you get an equation that contains only one variable. Sometimes you will have an equation that “contains no” variables. Double-check your solution to make sure you’ve replaced equation one (transformed) into equation two, not back to equation one. If you are confident that you will not make any mistakes, one of the following results will be obtained: ^{[1] X Source of Research}

If you have an equation that contains no variables and is incorrect (eg 3 = 5), then the problem has no solution . (If you graph the two equations, you’ll see that they’re parallel and never intersect.)
If you have an equation that contains no variables but holds true (for example, 3 = 3), then the problem has infinitely many solutions . These two equations are exactly the same. (If you graph the two equations, you’ll see that they are a single line.)

Using the suppression method

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Find the nullable variable. Sometimes equations can inherently “cancel” a variable when you add them together. For example, when you add up the two equations 3x + 2y = 11 and 5x – 2y = 13 , “+2y” and “-2y” cancel each other out, and remove all “y” variables from the equation. Look at the equations in your problem to see if there are any variables that can cancel out. If none of the variables can be suppressed, please refer to the next step.

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Multiply an equation to cancel a variable. (Skip this step if the variable cancels out.) If two equations with no variables can cancel themselves, transform one equation to do so. For example:

You have a system of equations 3x – y = 3 and -x + 2y = 4 .
Let’s transform the first equation so that the y variable cancels out. (You can choose the variable x and still end up with the same answer.)
The -y variable in the first equation needs to be suppressed with + 2y in the second equation. This can happen if you multiply -y by 2.
Multiply both sides of the first equation by 2: 2(3x – y)=2(3) , so 6x – 2y = 6 . Now – 2y will cancel out with +2y in the second equation.

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Add the two equations together. To add two equations, you add the left sides together, then add the right sides together. If you have properly transformed the first equation, one of the variables will cancel out. The following example uses the same equations as the last step:

You have the equations 6x – 2y = 6 and -x + 2y = 4 .
Add the two left sides: 6x – 2y – x + 2y = ?
Add the two right sides: 6x – 2y – x + 2y = 6 + 4 .

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Solve for the remaining variable. Simplify the added equation and then use basic algebra to find the final variable. ‘ If there are no variables left after the reduction, go to the last step in this section . Otherwise, you will get a result for one of the variables. For example:

You have 6x – 2y – x + 2y = 6 + 4 .
Group the variables x and y together: 6x – x – 2y + 2y = 6 + 4 .
Simplify the equation we have: 5x = 10
Solve for x: (5x)/5 = 10/5 , so x = 2 .

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Solve for the remaining variable. You’ve found a variable, but the job isn’t done yet. Substitute the result into one of the original equations so you can find the other variable. For example:

You know that x = 2 , and one of the two original equations is 3x – y = 3 .
Replace 2 in x: 3(2) – y = 3 .
Solve for y in this equation: 6 – y = 3
6 – y + y = 3 + y , so 6 = 3 + y
3 = y

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When two variables cancel, what to do? Sometimes adding two equations creates an equation that doesn’t make sense, or the equation just doesn’t help you solve the problem. Check the solution from the beginning, but if there are no errors, the answer will be one of two cases: ^{[2] X Research source}

If the equation adds up with no variables and is not true (like 2 = 7) then the system of two equations has no solution . (If you graph the two equations, you’ll see that they’re parallel and never intersect.)
If the equation adds up with no variables but is true (like 0 = 0) then there are infinitely many solutions . These two equations are actually exactly the same. (If you plot the two equations, you’ll see that they are a single line.)

Plot the equation

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Use this method only when required by the assignment. Unless you’re using a calculator or calculator with graphing capabilities, many systems of equations can only be roughly solved with this method. ^{[3] X Research Resources} Your teacher or textbook may ask you to use this method to get you used to graphing equations. You can also use the graphing method to check the answers of other solutions.

The basic idea is to graph the two equations and find their intersection. The x and y values at this point are the values of x and y in the system of equations.

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Solve both equations to find y. Separate the two equations and use algebra to convert the equations to “y = __x + __”. ^{[4] X Research Source} Example:

The first equation is 2x + y = 5 . Transform it to y = -2x + 5 .
The second equation is -3x + 6y = 0 . Transform it to 6y = 3x + 0 , then reduce it to y = ½x + 0 .
If both equations are identical then the entire line will be the “intersection”. Writing is infinitely many solutions .

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Draw the coordinate axes. On graph paper, you draw the vertical “y-axis” and the horizontal “x-axis”. Starting at the intersection of these two lines, you number 1, 2, 3, 4 etc.. going up on the y-axis and from left to right on the x-axis. Numbered -1, -2 etc.. from the origin going down on the y-axis, and from right to left on the x-axis.

If you don’t have graph paper, use a ruler to make sure the numbers are evenly spaced.
If you are using large numbers or decimals, you will need to scale the graph differently. (For example 10, 20, 30 or 0.1; 0.2; 0.3 but not 1, 2, 3).

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Determine the intersection of each graph with the y-axis. After the equation is converted to y = __x + __ , you begin to plot it by finding its intersection with the y-axis. The y value of the intersection is always equal to the last number in this equation.

In our original example, the line ( y = -2x + 5 ) intersects the y axis at 5 . The line ( y = ½x + 0 ) intersects the y axis at 0 . (These are the points (0.5) and (0,0) on the graph.)
If possible, you should use a different color pen to draw two lines.

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Use the slope to continue drawing the line. The form of the equation y = __x + __ has the number in front of x being the slope of the line. Every time x increases by one unit, the y value increases by the amount of the slope. Use this information to determine the point on the graph for each line when x = 1. (Or you can substitute x = 1 in each equation and solve for y.)

In the above example, the line y = -2x + 5 has a slope of -2 . At x = 1, the line moves down 2 units from the point with coordinates x = 0. Draw a line segment between the two points (0.5) and (1,3).
The line y = ½x + 0 has a slope of ½ . At x = 1, the line moves up ½ unit from the point with coordinates x = 0. Draw a line segment between the two points (0,0) and (1,½).
If two lines have the same slope , they never intersect, so the system of equations has no solution. Writing is futile .

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Continue plotting the lines until they intersect. Pause to observe the graph. If the two lines already intersect, go to the next step. Otherwise you will have to make a decision based on the situation of the two lines:

If two lines are moving towards each other then you continue to define more points in that direction.
If the two lines are moving away from each other, locate the points of the graph in the opposite direction, starting with x = -1.
If the two lines are very far apart, then you determine the very far points of the graph, for example x = 10.

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Find the solution of the system of equations at the intersection point. When two lines intersect, the x and y values there are solutions to the problem. If you’re lucky, the solution will be an integer. In the above example, the two lines intersect at (2,1) so the solution to the system of equations is x = 2 and y = 1 . With some systems of equations, the lines will intersect at the point between two integers, and you can hardly read the value unless your graph is extremely precise. When this is the case, you can write the answer as “x is between 1 and 2”, or use substitution or elimination to find the exact solution.

Advice

You can check your work by replacing your answers with the original equations. If the equations are correct (eg 3 = 3) then the answer is correct.
In the suppression method, you will sometimes have to multiply an equation by a negative number to cancel out a variable.

Warning

You cannot use these methods if the equation has a variable that contains an exponent, such as x ² . For more information on equations of this form, look up the literature on factoring quadratic equations. ^{[5] X Research Sources}

Steps

Use the alternative method

Using the suppression method

Plot the equation

Advice

Warning

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