How to Solve Systems of Equations Containing Two Variables

A system of equations has two or more equations with the same number of variables. To solve systems of equations containing two variables, you need to find an ordered pair that makes both equations true. It is simple to solve these equations by using the substitution method.

    Solve the system of equations, 2x+3y=1 and x-2y=4 by the substitution method.

    Take one of the equations from Step 1 and solve for either variable. Use x-2y=4 and solve for x by adding 2y to both sides of the equation to get that x=4+2y.

    Substitute this equation for x from Step 2 into the other equation 2x+3y=1. This then becomes 2(4+2y)+3y=1.

    Simplify the equation in Step 3 by using the distributive property and then adding like terms to get 8+7y=1. Now solve for y by subtracting 8 from both sides of the equation and the equation reduces to 7y= -7. Divide each side by 7 and y= -1.

    Find the value of the remaining variable x by using one of the equations in Step 1 and substituting y= -1. Let's choose x-2y=4 and substitute y= -1 to get that x+2=4. Then x is equal to 2 from this final equation and the ordered pair is 2,-1.

    Check this ordered pair in both of the original equations in Step 1 to verify that this is the solution.


    • You can also use the elimination, matrix or graphing methods to solve systems of equations containing two variables (see Resources below).

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