Section 5: Basic Algebra

An algebraic expression is a combination of numbers and variables connected by some mathematical operation, like addition, subtraction, multiplication or division. A variable is a letter (we often use x and y) that represents or is a holding place for a number. In statistical analysis, we will define variables and sometimes will use algebraic equations to relate two (or more) variables.

Examples of algebraic expressions: 2x + y, 5d, 10-r

Evaluating an Algebraic Expression
To evaluate an algebraic expression, replace the variable with the given number.

Example:
Evaluate 2x + 5 when x = 10.
Substitute 10 in for x: 2(10) + 5
Solve: 2*10 + 5 = 25 (Note: 2 * 10 is the same as 2(10).)

Equations
An equation is two expressions set equal to each other. For example, 2 + 2 = 4 is an equation. Equations can include variables (such as x and y). So, for another example, 3x – 5 = 10 is an equation (though it is only “true” for one value of x.)

The solution to an equation is the number, such that when you replace the variable, makes the equation true (i.e. the left side equals the right side.)

Example: Determine if any of these values of x is a solution to the following equation:
3x – 5 = 10
a) x = 5
b) x = -5

Substitute in the value for x (x = 5) and solve:
3(5) – 5 = ?
15 – 5 = 10 √

So, x = 5 is a solution because the left hand side equals 10 and the right hand side equals 10.

Now try x = -5: 3(-5) – 5 = ?
-15 – 5 = -20 χ

x = -5 is not a solution because the left hand side equals -20 and the right hand side equals 10.

General Solutions to a Linear Equation
Sometimes you will be given a linear equation and will need to solve for x (i.e. find a value for x that makes the equation true.) To solve linear equations, remember the following properties:

  • If you add or subtract a value from both sides of an equation, then the equation is still true. For example: If you have the equation 2x = 4 and add 5 to both sides (like this: 2x + 5 = 4 + 5) then the equation is still true.
  • If you multiply or divide on both sides of an equation by the same number (except 0), then the equation remains true. For example: If you have the equation 2x = 4 and divide both sides by 2 (like this: 2x ÷ 2 = 4 ÷ 2 or x = 2) then the equation will still be true.

You use these properties to solve a linear equation.

Example: Solve the following equation for x: x + 5 = 12

Subtract 5 from both sides:
x + 5 – 5 = 12 – 5
x = 7

So, x = 7 is the solution to this linear equation (i.e. 7 is the value for x that makes the equation true.)

Example: Solve the following equation for x: 3x – 4 = 8

Add 4 to both sides:
3x – 4 + 4 = 8 + 4
3x = 12

Divide both sides by 3 (to get x alone):
3x ÷ 3 = 12 ÷ 3
x = 4
Remember: if you divide by a fraction (e.g. 

image of 1/2

 ), it is the same thing as multiplying by the reciprocal or inverse of the fraction (e.g., 

image of 2/1

 or 2)

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