2.04 Linear Equations

Linear equation

An equation where the variable is raised to the highest power of 1, it will produce a straight line when plotted on a graph. Linear equations normally have the form of ax + b = c, where a, b, and c are constants, and a ≠ 0.

Inverse operation

Opposite mathematical operation that undoes another. For example, subtraction is the inverse operation of addition.

How do you solve linear equations?

1. Isolate the variable/subject you want to solve for by doing the inverse operations to both sides of the equation of operations that have already been performed on the variable.

   1. If the variable is on both sides, then rearrange the linear equation so terms containing the variable are all on 1 side of the equation, and the other terms are arranged the other side of the equation.

   2. If the variable has a fractional coefficient or is in the denominator → multiply the entire equation by the lowest common denominator to inverse the division of the denominator.

2. The order in which you perform these inverse operations are often the reverse of BIDMAS, (i.e. SAMDIB), however this depends on the order which the operations were applied to the variable to form the equation.

3. If you have time, substitute your answer back into the equation to check your answer is correct.

Example: Solve the equation 2/(x + 3) = 3/(4 - 2x)

2/(x + 3) = 3/(4 - 2x)

Multiply the entire equation by the lowest common denominator (x + 3) (4 - 2x)

2 (4 - 2x) = 3 (x + 3)

Expand the brackets and simplify.

8 - 4x = 3x + 9

Isolate the variable by performing inverse operations.

7x + 9 = 8

7x = -1

x = -1/7

How should you layout your equations and solutions?

1. Align the equal signs on each line.

2. Arithmetic operations must be performed to BOTH sides of the equation, 1 operation at a time.

3. Working out should be done downwards, NOT horizontally.

4. ALWAYS keep the variable(s) on the same side of the equation.