How to Balance Linear Equations

The equals sign separates the two sides of an equation that must be kept balanced.

Solving an algebraic equation entails isolating a variable, or unknown, so that you can find its numerical value. However, during this process you must keep the equation balanced at all times. This is because an equation is like a seesaw---for the equals statement to be true, neither side can be bigger (higher) or smaller (lower) than the other. If it becomes unbalanced, it will no longer be equal and will, therefore, no longer be true. Hence, each step to solving an equation also works to keep it balanced.

Instructions

1. Use the addition property of equality to balance the equation while changing it so that the variable, represented by a letter, is only on one side. The property states that you can keep the equation balanced by adding the same number to both sides. For instance, if you have 7x - 2 = 4x + 5, you would add - 4x to both sides to get 3x - 2 = 5, and if you had 7x - 2 = -4x + 5, you would add 4x to both sides to get 11x - 2 = 5.

2. Look at the equation and find the plain number, or constant, that is on the same side as the variable. For instance, in 11x - 2 = 5 it would be -2 because it is on the same side as the x. Then, add the opposite of that number to both sides to keep the equation balanced while further isolating the variable. Here, you would add 2 to both sides and end up with 11x = 7.

3. Use the division property of equality to get rid of the coefficient in front of the variable by dividing both sides by the same number. For instance, you would divide both sides of 11x = 7 by 11 to get x = 7/11. If the variable has a fraction coefficient, as in 2/3x = 6, you would use the multiplication property of equality and multiply both sides by the reciprocal of the fraction. Here, both sides would be multiplied by 3/2 to get x = 9. At this point, you have solved the equation because the variable is alone. Furthermore, the answer is correct because you have kept the equation balanced.

Tips & Warnings

• If there are variables on both sides, distribute any numbers outside parentheses before you begin solving and balancing. For example, in 3(2x - 6) = 12x + 4, you would distribute the 3 to get 6x - 18 = 12x + 4. The same is true if there is a division bar; if you had (2x - 6)/3 = 12x + 4, you would have to multiply each thing in the parentheses by 1/3 before beginning.

• If the variable is only on one side, do not distribute the number in front of the parentheses, just divide both sides by that number. For example in 3(2x - 6) = 4, you would divide both sides by 3 to get 2x - 6 = 4/3. The same goes for division; you would multiply both sides by the number under the division bar.

• Remember that when you add something to both sides, you only have to do it in one spot to keep the equation balanced, but when you multiply or divide, you must do it to every term on each side.

References

• Mathwords: Properties of Equality
• Study Guides and Strategies: Linear Equations