How to Use Simultaneous Equations

In any algebraic system of equations, the number of variables that can be solved for is dependent on the number of equations. One variable (most often X) requires at least one equation; solving for three variables (X, Y and Z) requires at least three equations. Using more than one equation at a time to solve for more than one variable is called using simultaneous equations and is relatively easy.

Difficulty: Moderate

Instructions

Things You'll Need:

• At least two unknowns (X and Y in this example)
• At least two equations

1. Write both equations on the same sheet of paper.

2. Rearrange both equations so that both unknowns are on the left side of the equation and all constants are on the right side.

3. Rearrange both equations if necessary so that the two variables are in the same order in each equation. For example, if the first equation is "4X + 2Y = 34," the second equation should read "19X -- 4Y = 121" rather than "-4Y + 19X = 121."

4. Choose an unknown to solve for. In this example, we will solve for X.

5. Manipulate one or both equations so that the coefficients of the Y variables are opposites. In this example, multiplying the first equation by 2 will result in "8X + 4Y = 68," resulting in "4" and "-4" for the Y coefficients. You may have to add, subtract, multiply or divide one of the equations by some constant in order to do this.

6. Place one equation on top of the other, and add the individual terms. In other words, add the X coefficients, the Y coefficients and the constants to equal a new, third equation. In this example, 8X + 19X = 27X, 4Y + (-4)Y = 0, and 68 + 121 = 189. This results in a new equation of 27X = 189.

7. Solve for X. In this example, dividing both sides by 27 yields X = 7.

8. Substitute this value for X in one of the original equations. This will yield a new equation with only one variable: Y.

9. Solve for Y in the new equation. In this example, if you substitute 7 for X in "4X + 2Y = 34" and solve for Y, you get Y = 3.

Tips & Warnings

• The coefficient is the number next to the variable. For example, in the first equation in Step 3, the coefficient of X is 4 and the coefficient of Y is 2. Watch your signs! Remember that subtracting a negative number is equivalent to adding, and that if you subtract a larger number from a smaller, you will end up with a negative number. Remember that in an equation, everything you do to one term in the equation must be done to all terms in the equation. If you multiply the Y coefficient by 3, you must also multiply the X coefficient and the constant by 3.

References

• Elementary and Intermediate Algebra, 3rd Edition; Jerome E. Kaufmann; 2002