How to Solve a System of Linear Equations With Answers

Isolate variables in the equations of a system to solve for their values.

A system of linear equations contains two variables (usually x and y) and two equations of those variables. The solution to a system of linear equations is the values of x and y that make both equations true. Some systems of equation have no answers -- this means no value of x and y makes both equations simultaneously true. Substitution is a straightforward method that will always find the solution of a system of equations with answers.

Instructions

1. Choose one of the equations and combine all like terms (like terms are constants and terms with the same variable). For example, in the equation 2x - y = 4 - 3y, you could combine the like terms "-y" and "-3y" by adding y to both sides of the equation to get the equation 2x = 4 - 2y.

2. Isolate one of the variables on one side of the equation by subtracting any other terms from both sides and then dividing by the coefficient of x. In the equation 2x = 4 - 2y, you would divide both sides of the equation by 2 to isolate x: x = (4 - 2y) / 2 or x = 2 - y.

3. Substitute the value of this variable into the other equation. For example, if the other equation is x - y = 6, substitute "2 - y" in for x to get the equation 2 - y - y = 6.

4. Solve this equation by isolating the one variable. Isolating y in the equation from Step 4 yields the solution -2y = 8 or y = -4.

5. Plug the solution of the variable in Step 4 into the simplified equation from Step 2 to solve for other variable. Plugging -4 in for y in the equation x = 2 - y yields the solution x = 2 - (-4) or x = -2. The solution to the system of equations is the value of the two variables.

References

• Purple Math: Systems of Linear Equations -- Solving by Substitution