Solving Basic Math Equations Study Guide

Updated on Oct 3, 2011

Introduction to Solving Basic Math Equations

A problem well stated is half-solved.

—John Dewey (1859–1952)

Now that you have the major elements of equations mastered, this lesson will show you how to solve equations using isolations, distribution, and factoring.

The main thing you have to know about solving equations is this—what you do to something on one side of an equation you must do to the other side of the equation.

In order to keep the meaning of the original equation, make sure that you are doing the same thing to both sides of the equation. This means that you should perform corresponding operations on both sides of the equal sign. If you subtract 2 from the left side, you need to subtract 2 from the right side. If you divide the left side by 3, you must divide the right side by 3.

An equation is officially solved when the variable is alone (isolated) on one side of the equation and the variable is positive. Any equation with one variable can easily be solved by changing the equation this way. When you want to solve for the variable, you want to get it all by itself. This is called isolating the variable. To do this, you can add, subtract, multiply, or divide both sides of the equation by the same number. Let's try to isolate the variable x in the following equation:

25 –2x = 35

Subtract 25 from both sides.

Try to divide both sides by 2.

–x = 5

Remember, though, the variable must be positive, so –x = 5 is not the answer. Remember that –x is the same as –1x. In order to make the variable positive, divide both sides by –1.

x = –5

Let's try another one.

2x + 7 = 15

You have 2x + 7 = 15. In order to get x by itself, first get rid of the 7. Subtract 7 from both sides.

Now, divide both sides by 2 in order to get x by itself.

x = 4

Distributing and Factoring

The distributive law states that the sum of two addends multiplied by a number is the sum of the product of each addend and the number.

a(b + c) = ab + ac


An addend is any number to be added.

Let's look at an example. What is the value of x if 12(66) + 12(24) = x? Using the distributive law, x equals 12(66 + 24), or 12(90). This simplifies to 1,080.

When you use the distributive law to rewrite the expression ab + ac in the form a(b + c), you are factoring the original expression. In other words, you take the factor common to both terms of the original expression (a) and pull it out. This gives you a new "factored" version of the expression you began with.

When you use the distributive law to rewrite the expression a(b + c) in the form ab + ac, you are unfactoring the original expression.

Find practice problems and solutions for these concepts at Solving Basic Math Equations Practice Questions.

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