How to Solve Simultaneous Equations Using Substitution

Algebraic equations with more than one unknown are difficult to solve. When you have more than one equation with the same unknowns, substituting values from one equation into the next is an easy way to solve one of the unknowns. Then you can use that value to solve the second unknown.

Difficulty: Moderate

Instructions

1. Choose two equations with the same X and Y unknowns. For example, use 2X + Y = 4 and X - Y = -1.
   

2. Select one of the equations and one unknown to solve. For example, use 2X + Y = 4 and solve Y.

3. Isolate Y on the left side of the equal sign using algebraic arithmetic to move all other terms to the right of the equal sign. Rewrite the first equation to look like Y = 4 - 2X.

4. Rewrite the second equation replacing Y with the equivalent expression from the right of the first equation to form an equation with X as the only unknown. The second equation now looks like X - (4 - 2X) = -1
   

5. Solve the new equation in Step 4 for the single unknown using standard algebraic methods. To solve the example, rewrite the equation to look like 3X = -1 + 4. Then solve 3X = 3 so X = 1.
   

6. Replace the value of X from Step 5 in the first equation in Step 3 to solve for Y. Write Y = 4 – 2(1) and solve to discover Y = 2.