## Linear Equations A linear equation is a mathematical statement containing one variable.

To solve an equation, the value or values of the variable must be found that makes the statement true.

There are several types of equations and several methods of solving linear equations.

When solving equations each step should be written on a new line, and the equals signs should be kept directly underneath each other.

### Simple Equations

• The aim when solving an equation is to get one of the variables on its own, on one side of the equation.
• To remove a term from one side of an equation, carry out the opposite operation to both sides of the equation.
• The answer should then be checked by substituting it back into the equation to make sure that both sides have the same value. This step can be done mentally.

 Examples Answers Solve: x + 7 = 15 x + 7 = 15 x + 7 − 7 = 15 − 7 (subtract 7 from both sides) x = 8 g − 3 = 12 g − 3 = 12 g − 3 + 3 = 12 + 3 (add 3 to both sides) g = 15 3w = 15 3w = 15 3w⁄3 = 15⁄3 (divide both sides by 3) w = 5 k⁄4 = 5 k⁄4 = 5 k⁄4 × 4 = 5 × 4 (multiply both sides by 4) k = 20

### Equations with Brackets

Generally, it is better to expand any brackets first.

 3(x + 7) = 24 3x + 21 = 24 expanding brackets 3x + 21 − 21 = 24 − 21 subtracting 21 from both sides 3x = 3 simplifying x = 1

### Equations with Fractions

There are two ways to deal with fractions:

1. Multiply both sides by the reciprocal. 2. Multipy every term by the common denominator.

 Example 1 Example 2  ### Equations with Two Variable Terms

If the variable is on both sides of the equation, collect the variable terms on the side that has the most.

 3x + 7 = 2x + 8 3x − 2x + 7 = 2x + 8 − 2x (subtracting 2x from both sides) x + 7 = 8 (simplifying) x + 7 − 7 = 8 − 7 (subtracting 7 from both sides) x = 1 (simplifying)

### Special Equations

• Some equations have no solution.

e.g. x + 3 = x + 4

x = { } (the empty set)

• Some equations have an infinite number of solutions.

e.g. 3(x + 4) = 3x + 12

3x + 12 = 3x + 12
x can have any value.

### Problem Solving

Many problems can be solved by converting them to algebraic equations and solving them.

The general way to approach these problems is:

• Assign a variable to the unknown quantity in the problem.
• Make up an equation from the information in the problem.
• Solve the equation.

 Example Answer Hemi thinks of a number, doubles it and subtracts 8. The result is 32. What is his number? Let the unknown number be x. The equation is: 2x − 8 = 32 Solving the equation: Hemi's number is 20