Solving Equations

Learning Objective(s)

·         Isolate variables using the Inverse Property or Inverse Operations.

·         Solve algebraic equations using the Properties of Equality.

Introduction

Math is all about equations. Numerical equations give us a precise way to describe, share, and solve problems. Algebraic equations let us explore even further by giving us a way to tackle problems that include unknown quantities.

In order to succeed in algebra, you must understand what equations are and how to write and solve them.

Parts of an Equation

An equation is a mathematical statement that two expressions are equal. In a simple numerical equation, expressions made of numbers and operations appear on each side of an equals sign. The equals sign means that the two expressions have the same value. For example, 3 + 9 = 12 is an equation. The expression on the left, 3 + 9, has the same value as the expression on the right, 12. They are written in different ways, but both represent the same amount.

Algebraic equations are made of more than just numbers—they also contain variables, symbols that stand for an unknown quantity. Variables are usually letters, like x, y, or z. Sometimes a variable will be multiplied by a number. This number is called the coefficient of the variable.

In the equation 2y = 14, what is the variable and what is the coefficient?

A) The variable is y and the coefficient is 2.

B) The variable is y and the coefficient is 14.

C) The variable is 14 and the coefficient is 2.

D) The variable is 2 and the coefficient is y.

Solving Equations

Equations are not fixed—they can be rearranged, as long the equality of the two sides is maintained. We rewrite equations in order to make them simpler, or to solve them.

Solving algebraic equations often means isolating the variable, or rewriting the equation so that one expression is only the variable with a coefficient of 1. To get a variable alone, we have to find a way to shift the rest of its expression and the original coefficient to the other side of the equation.

Isolating Variables with the Inverse Property

There are two ways to get a variable alone using the Inverse Property.

The Additive Inverse Property says that any number added to its opposite equals zero. We can use this property to get rid of any values around the variable term.

For example, in the expression y + 2, we can remove the 2 by adding its opposite, -2.  y + 2 + (-2) is y + 0. That’s just y.  We’ve cleared a number from the expression by adding its opposite.

The Multiplicative Inverse Property says that when we multiply any number by its inverse, the result is 1.

Consider the expression 4y. 4 is the coefficient of the variable y. We want the variable to have a coefficient of 1. So let’s multiply 4y by the multiplicative inverse of 4. That number is ¼. When we multiply ¼ times 4y the 4’s cancel out, leaving just 1y. We’ve turned the coefficient of y into 1 by multiplying the coefficient by its multiplicative inverse.

Isolating Variables with Inverse Operations

There’s a slightly different way to do the same thing—to clear away terms and reduce coefficients to 1—and that’s by carrying out Inverse Operations. Some people find this idea easier to work with than the Inverse Property. Inverse operations come in pairs—one operation undoes the other.  A non-math example of inverse operations is tying and untying your shoelaces. Tying your shoe creates a simple knot; untying your laces undoes this knot.

For example, consider the expression 9 + 8. We can “undo” the addition of 8 by carrying out the inverse operation and subtracting 8.  9 + 8 – 8 is just 9. The addition and subtraction of 8 have canceled each other out.

Multiplication and division are also inverse operations. Multiplication can be undone by division, and division can be undone by multiplication. In the expression 6x, x is multiplied by six. We can undo the multiplication by dividing by 6. 6x divided by 6 is just x.

How would you isolate the variable in the expression?

A) Divide by 3.

B) Multiply by.

C) Multiply by 3.

D) Divide by n.

Solving Equations with the Properties of Equality

The Inverse Property and Inverse Operations are only half the story in rearranging equations. Remember, the definition of an equation is that it is a statement of the equality of two expressions.  One way to think about an equation is to imagine the expressions that make it up sitting on each side of a balance scale.  Because the expressions have the same value, the scale is balanced.

When we add, multiply, subtract, or divide one of those expressions, we change its value. If that’s all we do, we’ve unbalanced the scale—the two sides of the equation are no longer equal.

Lucky for us, the Properties of Equality explain how we can keep the scale and the equation in balance. Whenever we do an operation on one side of an equation, if we also do the exact same thing to the other side, we keep both sides equal.

Let’s see how this works with a simple numeric equation, ½(10) − 1 = 4. Let’s get rid of the -1 on the left side of the equation by adding its opposite, +1.

We need to balance this change by doing the exact same thing to the right side of the equation. So we must also add +1 to 4.

Because -1 + 1 = 0, the expression on the left becomes just ½(10). On the right side of the equation, 4 +1 equals 5, so that expression is now 5. Notice that the two sides of the equation still balance—they are still equal to one another.

How would you solve the equation 5 + 7z = 19?

A) Subtract 5 from both sides of the equation and then divide both sides by 7.

B) Divide both expressions by 7 and then add -5 to each side of the equation.

C) Add -5 to the left side of the equation and then multiply by 1/7.

D) Add 5 to both sides of the equation and then multiply each side by 7.

Summary

Equations are mathematical statements that combine two expressions of equal value.

There are several techniques for solving algebraic equations, which are equations with variables. They all have the same goal—isolating a variable on one side of an equation by using the properties of equality and inverse to shift all the other terms to the other side of the equation.