The system of equations is solved using the linear combination method

A “system of equations” is type of math problem in which you have two or more separate equations and you need to find the values of two or more variables. In general, to be able to find a solution, you need to have as many different equations as the number of variables that you wish to find. (There are advanced problems where the number of equations and the number of variables do not match, but that will not be addressed here.)

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   Recognize the standard format. In algebra, the "standard format" for an equation is one that is written as

   
   .[1] When written in this format, the letters A, B and C are commonly chosen to represent numerical values, while x and y are the variables that you need to solve.

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   Rearrange your equations to put them into standard format. This may require you to combine similar terms, if each variable appears in the equation more than once, for example.[2] You will also need to move the terms so they appear in the proper order.[3]

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   3

   Write your equations so the variables line up. It is helpful to write your equations with one directly over the other, so the similar terms line up.

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   1

   Examine the equations in standard format. When you have your equations written in standard format, lined up so the similar terms are aligned, check the coefficients. You are looking for one pair of coefficients that match.[4]

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   2

   Subtract corresponding terms. Working across the system from left to right, subtract each term of the second equation from the corresponding term of the first equation.

   • It may be helpful simply to draw a long horizontal line across the bottom of the two equations and subtract downward, as you would with any ordinary subtraction problem.

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   3

   Write out the result. If one of your terms matched exactly, as it should, and you subtracted correctly, then one of the variables should be eliminated from the problem. Rewrite what you have left as a single equation.

   • In the example above, you should be left with .
   • Because one of the variables gets eliminated in this method, some textbooks will refer to this as the "elimination" method of solving a system of equations.

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   Solve for the remaining variable. What you have left should be a fairly simple, one-variable equation. Solve it by dividing both sides of the equation by the coefficient.[5]

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   Replace that solution into one of your original equations. Take that solution, in our example y=1, and substitute it in place of in either one of the original equations.

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   Solve for the remaining variable. Use basic algebraic steps to solve for the remaining variable. Remember that whatever action you do to one side of the equation, you must also do to the other side.[6] For example:

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   7

   Check your two solutions. Verify that you have done the work correctly by checking your solutions. You should be able to place your two solutions, in this example and , into each of the original equations. When you then simplify the equations, you will get true statements.

   • For example, check the first equation as follows:
   • Check the second equation as follows:

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   Write out your solution. The final solution, which you have proven to work in both equations, is and .[7]

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   1

   Examine the equations in standard format. Set up your two equations in standard format and look at the coefficients of each of your variables. You are looking for the circumstance where the numbers are the same but the signs are different.[8]

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   2

   Add corresponding terms. Working across the system from left to right, add each term of the first equation to the corresponding term of the second equation. It may be helpful simply to draw a long horizontal line across the bottom of the two equations and add downward, as you would with any ordinary addition problem.

   • The above example works out as follows:

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   3

   Write out the result. Because you were adding, and one of your terms contained opposites, then one of the variables should be eliminated from the problem. Rewrite what you have left as a single equation.

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   4

   Solve for the remaining variable. What you have left should be a fairly simple, one-variable equation. Solve it by dividing both sides of the equation by the coefficient.

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   5

   Solve the second variable. Take that solution, in our example x=8, and substitute it in place of in either one of the original equations.

   • Choose the first equation:

6. 

   6

   Check your two solutions. Verify that you have done the work correctly by checking your solutions. You should be able to place your two solutions, in this example and , into each of the original equations. When you then simplify the equations, you will get true statements.

   • For example, start with the first equation:
   • Now try the second equation:

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   Write out your solution. The final solution, which you have proven to work in both equations, is and .[9]

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   1

   Examine the equations in standard format. It is more likely that your system of equations will not have a pair of matching or opposite coefficients. When you line up the two equations and compare coefficients, unless two coefficients (the A and B of the standard format) match exactly, you need to take a couple extra steps.[10]

   • For example, consider these two initial equations:
   • When you examine them, there are no matching coefficients for similar terms. That is, the 3x does not match the 8x, and the 2y does not match the -4y. There is also no pair of opposites.

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   2

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   Either add or subtract the two new equations. If you have created a matching pair of coefficients, you will subtract terms to eliminate one variable. If you have created a pair of opposite coefficients, you will add terms to eliminate one variable. Consider the following example:

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   4

   Replace that solution into one of your original equations. Take that solution, in our example x=1, and substitute it in place of in either one of the original equations. This works as follows:

5. 

   5

   Check your two solutions. Verify that you have done the work correctly by checking your solutions. You should be able to place your two solutions, in this example and , into each of the original equations. When you then simplify the equations, you should get true statements.

   • For example, check the first equation:
   • Now check the second equation, as follows:

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   6

   Write out your solution. The final solution, which you have proven to work in both equations, is and .[11]

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   1

   Recognize identical equations as having infinite solutions.[12] In some circumstances, your system of linear equations may have infinite solutions. This means that any pair of values that you insert into the two variables will make the two equations correct. This happens when the two equations are really just algebraic variations of the same, single equation.

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   Find systems with no solution.[13] Occasionally you may have a system in which the two equations, when written in standard form, are nearly identical except that the constant term C is different. Such a system has no solution.

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   Use a matrix for systems with more than two variables.[14] It is possible for a system of linear equations to have more than two variables. You may have 3, 4, or as many variables as the problem dictates. Finding a solution to the system means finding a single value for each variable that makes each equation in the system correct. To find a single, unique solution, you must have as many equations as you have variables. Thus, if you have the variables and , you need three equations.

   • Solving a system of three or more variables can be done using the linear combinations explained here, but that gets very complicated. The preferred method is using matrices, which is too advanced for this article. You may wish to read Use a Graphing Calculator to Solve a System of Equations.

Add New Question

• Question

  Given the two equations 6x+ 2y = 2 and 8x + 3y = 14, explain how knowing how to find the least common multiple (LCM) of two numbers can help you solve the system of equations presented here by eliminating the x-terms.

  

  These equations are already written in standard format, Ax+By=C. However, the x coefficients do not match, so they are not ready to be eliminated. To eliminate the x terms, you need to do a multiplication step. The quickest way to multiply would be to multiply the first equation by the x-coefficient from the second equation, and multiply the second equation by the x-coefficient from the first equation. This would give you the equations 48x+16y=16 and 48x+18y=84. These are very large, bulky numbers. If you recognize that 6 and 8 have a least common multiple of 24, you could choose different numbers to multiply. Multiply the first equation by 4 and the second equation by 3, and you will create 24x+8y=8 and 24x+9y=42. You can now solve the system more easily.

• Question

  How do i solve 2×+5y=1/3×-2y=8

  

  This is a strange way of writing the problem. By using two equals signs in a line like this, you have combined two different equations. Breaking these apart gives 2x+5y=8 and 1/3 x -2y=8. Because each equation equals 8, you can just drop the 8 and put the two equations as equal to each other, so you have 2x+5y=1/3 x-2y. Because you have two variables here, and only one equation, you cannot get a single solution. The best you can do is simplify the equation by combining like terms. This will give 1 2/3 x = -7y. If you wish to write this in a slope-intercept form to describe a line, you could divide both sides by 7, and switch sides, to get y= -5/21 x. My guess is that you probably copied your problem incorrectly. Check it again.

• Question

  This is just the elimination method. Is there any other simpler method?

  

  There's also the substitution method, but it is only simpler in some cases.

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• Note that linear combinations is only one way to solve a system of equations. You can also solve a system by substitution. For help on that, see Solve Systems of Algebraic Equations Containing Two Variables.

• Always pay careful attention to the signs of the coefficients. Common mistakes occur when people miss negative signs. Remember that if the signs match, then you will subtract terms to eliminate a variable. If the signs are opposites, then you will add to eliminate a variable.

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