Linear Equations

A linear equation is an equation in which the variable (usually x or y) is raised only to the power of one — never squared, cubed or under a root sign. Examples: x + 5 = 12, 3y − 7 = 8, 2x + 3y = 12. The word linear comes from the fact that, when graphed, every linear equation produces a straight line.

Linear equations are everywhere in real life: working out how many minutes of M-Pesa airtime you can afford, calculating distances with rate × time, sharing a bill between friends. The CBC Mathematics syllabus from Grade 7 onwards spends a lot of time on this skill because it is the foundation for every other algebraic technique that follows.

Standard form of a linear equation:

• One variable: ax + b = c (where a, b, c are constants and x is the variable). Example: 3x − 5 = 7.
• Two variables: ax + by = c. Example: 2x + 3y = 12.

The balance method (one variable):

Treat the equation as a balance scale. Whatever you do to one side, you MUST do to the other to keep it balanced. The aim is to isolate the variable on one side.

Step-by-step rules:

1. Remove brackets by expanding (distributive property).
2. Combine like terms on each side.
3. Add or subtract the same number from both sides to move constants to one side.
4. Multiply or divide both sides by the same number to isolate the variable.
5. Write the answer and check by substituting back.

Solving simultaneous equations (two variables):

When you have two equations in two unknowns, you need both to find unique values for x and y. Three common methods:

• Substitution — solve one equation for one variable, then substitute into the other.
• Elimination — add or subtract the two equations to eliminate one variable.
• Graphical — draw both lines on a graph and find the intersection point.

Word problems — three-step pattern:

1. Define the variable. "Let x = the number of mangoes."
2. Form the equation. Translate the word problem into algebra.
3. Solve and interpret in the context of the question.

Common mistakes to avoid:

• Sign errors when moving terms. x + 5 = 12 becomes x = 12 − 5, not x = 12 + 5. Always change the sign when crossing the equals sign.
• Dividing only part of one side. In 2x + 4 = 10, you cannot just divide everything by 2 — you must subtract 4 first (so 2x = 6), then divide (x = 3).
• Forgetting to apply the operation to BOTH sides. Whatever you subtract from the left, you MUST subtract from the right.
• Treating 2x as 2 + x. 2x means 2 multiplied by x. It is x + x, not x + 2.
• Not checking the answer. Substitute your answer back into the original equation. If both sides match, you're correct.

CBC Grade 4–5 introduces the use of letters in number sentences; Grade 6 covers simple linear equations with one variable; Grade 7 covers linear equations and linear inequalities in one variable, balance method, and word problems; Grade 8 introduces simultaneous equations in two variables; Grade 9 extends to graphical solutions and the equation of a straight line — material that appears in KJSEA and senior school Mathematics.