How to Solve a System of Two Equations Algebraically

A system of equations is a set of two or more equations that you solve at the same time. A system of linear equations is the easiest to solve, involving two equations and two variables, typically x and y. The simplest way to solve a system of equations is by the substitution method, which requires you to solve an equation for a variable, which you plug into the other equation in order to reach a solution.

Instructions

1. Determine whether you want to solve for x or y in your system of equations. For the purpose of the exercise, use the equations 2x-y = 9 and 3x-4y = -24.

2. Eliminate the x in the first equation, 2x-y = 9, by solving for y. Subtract 2x from each side of the equation to get -y = 9-2x. Make y positive by negating each term on the opposite side of the equation, leaving y = 2x-9.

3. Substitute 2x-9 for y in the second equation, which is 3x-4y = -24. After substituting for y, the equation is 3x-4(2x-9) = -24.

4. Multiply the 4 by (2x-9) to simplify the equation, leaving 3x-8x-36 = -24. Combine the terms of x before solving the equation, resulting in -5x+36 = -24.

5. Solve for x. Subtract 36 from each side to get -5x = -60. Divide each side by -5 to get x = 12.

6. Go back to the first simplified equation, y = 2x-9, and substitute 12 for x. The equation is y = 2(12)-9, which simplifies to y = 24-9. Subtract 24-9 to get y = 15 for the final answer.

References

• "Precalculus: Functions and Graphs"; Mustafa A Munem, et al.; 2002
• "Introductory Algebra: An Applied Approach: Student Support Edition"; Richard N. Aufmann, et al.; 2009