Variables and Solving Equations

Variables are letters or symbols that represent numerical quantities. For example, in the area formula for a square, A = s2, A represents the area and s represents the side length of the square. The variable x is often used to represent an unknown quantity. Also, y, z, and t are used a lot as variables.

An equation consists of the equating of two numerical or algebraic expressions with an “=” sign.

Examples of Equations

x + 4 = 18

Here, an unknown quantity x , when added to 4, results in 18.

3x = 12

Here, an unknown quantity x, multiplied by 3, results in 12.

Note that when a number is next to a variable, this implies multiplication.

4x + 1 = 10

Here, an unknown quantity x is multiplied by 4 and then one is added and the result is equal to 10.

Methods of Solving Equations

To solve an equation means to find the value or values of x that make the equation true. There are two ways to solve equations:

Method 1 – Solve and equation by inspection (guess and check).

Method 2 – Use algebraic properties of equality to solve for x.

CONTINUED ON NEXT PAGE

Example: Solve x + 4 = 18 by inspection and also by using algebraic properties of equality.

By inspection, we think to ourselves “What number added to 4 results in 18 ? ” The answer we quickly come up with is that x must equal 14. So the solution is x = 14. And if we plug in x=14, we get 14 + 4 = 18, a true statement.

By using algebraic properties, we subtract 4 from both sides of the equation to get

x + 4 – 4 = 18 – 4 and this simplifies to

x + 0 = 14 which simplifies to

x = 14

In this example we used the Subtraction Property of Equality, which states that you may subtract equal amounts from both sides of an equation without changing the equation.

Also, we used the Zero Property of Addition, which assures us that x + 0 simplifies to x.

Example: Solve 3x = 12 by inspection and also by using algebraic properties of equality.

By inspection, we think to ourselves “What number times 3 results in 12 ? ” The answer we quickly come up with is that x must equal 4. So the solution is x = 4. We verify that this is the solution by replacing x with 4 to get 3  4 = 12, a true statement.

By using algebraic properties, we divide both sides of the equation by 3 to get

3x = 12 which simplifies to

1x = 4 which simplifies to x = 4

In this example we used the Division Property of Equality, which states that you may divide both sides of an equation by equal amounts without changing the equation. Also, we used the Multiplication Property of One, which assures us that 1x simplifies to x.

Example: Solve 4x + 1 = 10 by inspection and also by using algebraic properties of equality.

By inspection, we think to ourselves “What number multiplied by 4 and then increased by 1 results in 10 ? ” Here, the answer is not very obvious. In fact, unless you perform algebraic operations mentally, it is very difficult to “guess and check” the answer. So let’s move on to the algebraic solution.

By using algebraic properties, we subtract 1 from both sides of the equation to get

4x + 1 – 1 = 10 – 1 which simplifies to

4x + 0 = 9 or 4x = 9 and then we divide both sides by 4 to get

4x = 9 which simplifies to

1x = 9/4 or x = 9/4

If we plug this back in, we get 4(9/4) + 1 = 10 or 9 + 1 = 10.

Here, we used both the Subtraction Property of Equality and the Division Property of Equality. Also, we used the Zero Property of Addition and the Multiplication Property of One.