Solving Equations

Understanding Equations

An equation represents a balance between two expressions; this balance is indicated by the equals sign (=).
The goal when solving an equation is to find the value(s) of the variable(s) that make the equation true.

The principle of balance is key in solving equations. Whatever you do to one side of the equation, you must do to the other side to keep the balance.
Addition and subtraction can be used to remove terms from one side of the equation.
Multiplication and division can be used to remove coefficients or deal with fractions in the equation.

A linear equation is an equation where the variable has no exponent other than 1. It appears in a form like ax + b = c.
To solve linear equations, the goal is to isolate the variable on one side, forming x = d (or other comparable forms).

Begin by eliminating any brackets using the distributive property a(b + c) = ab + ac.
If like terms on one side of the equation can be combined, do this to simplify the equation.
Use addition or subtraction to remove any additional terms from the side with the variable.
Finally, use division or multiplication to isolate the variable.

A quadratic equation is an equation that can be rearranged in standard form as ax² + bx + c = 0.
Quadratic equations can be solved by factorising, completing the square, or using the quadratic formula.
The solutions of a quadratic equation are called roots.

Forgetting to maintain balance can lead to incorrect solutions. Remember the golden rule: whatever is done to one side of the equation must also be done to the other side.
Misinterpreting the equals sign as an operation to perform rather than a symbol of balance is a common misunderstanding.
Be careful when solving equations with fractions, it’s often easier to eliminate the fractions first before solving the equation.
Forgetting that a quadratic equation can have two solutions. Always check to see if there is an additional valid solution.