How to Solve Equations Using More Than One Variable

By Jacob Nomi
Only terms with matching variables and exponents can be added.
Jupiterimages/Photos.com/Getty Images

You can solve equations with more than one variable by employing the substitution process. First, you isolate one of the variables in one equation and solve it in terms of the other variable. Then, you substitute that value into the other equation.

Solve for x and y given the following two equations: 1) 3x + y = 10; 2) -4x - 2y = 2. The first step is to isolate a variable and get a solution for that variable. In equation 1, y can be isolated by adding (-3x) to both sides of the equation: -3x + 3x + y = -3x + 10, resulting in the first answer: y = -3x + 10.

This new value of y can be used in equation 2) -4x - 2(-3x + 10) = 2. Perform the multiplication to begin solving for x: -4x + 6x -20 = 2. Adding -4x + 6x = 2x results in: 2x -20 = 2. The next step is to isolate the variable. First add 20 to either side of the equation: 2x - 20 + 20 = 2 + 20. Then multiply both sides of the equation by 1/2: (1/2)2x = (1/2)22.

x = 22/2 or 11.

You can solve for y using either equation 1 or 2. In equation 2 ( -4x - 2y = 2), you substitute the value of x to get: -4(11) - 2y = 2. This yields: -44 - 2y = 2. Next add +44 to both sides to isolate the y variable: +44 - 44 - 2y = 2 + 44, resulting in -2y = 46. If you multiply both sides of this equation by (- 1/2) you get the value of y: (- 1/2) -2y = (- 1/2) 46. Therefore the value of y is -46/2 or -23.

The solution to our set of equations is x = 11, y = -23 but you should always check to make sure your answer is correct. If you plug these values into either of the original equations, the equation should work: 1) 3x + y = 10 become 3(11) -23 = 10, or 33 - 23 = 10. Now try equation 2) -4x - 2y = 2: -4(11) -2(-23) = 2, or -44 + 46 = 2. The solution is correct.

About the Author

Jacob Nomi has been writing professionally since 2011. His areas of expertise include linguistics, law, Russian literature, exercise science and nutrition. Nomi holds a Master of Arts in Russian literature and linguistics from the State University of New York at Stony Brook.