Algebra is the part of maths where letters stand for numbers. This guide explains variables, equations and formulas, with worked examples you can check on the CalcSolver tools.
What is algebra?
Algebra uses letters, called variables, to stand for unknown numbers. Instead of working out 3 + 5 = 8, algebra lets you solve 3x + 5 = 20 and find that x = 5. The same rules work for any numbers, which is why algebra is used in science, engineering, finance and everyday problem solving.
Variables and expressions
A variable is a letter that stands for a number, such as x or y. An expression combines variables and numbers, for example 2x + 3. A term is one part of an expression, such as 2x or 3. The coefficient is the number in front of a variable: in 2x the coefficient is 2. Like terms have the same variable, so 3x + 5x = 8x, but 3x + 5y cannot be simplified further.
Solving linear equations
A linear equation has the variable to the power of 1, such as 3x + 5 = 20. The golden rule is: do the same thing to both sides. First undo addition and subtraction, then undo multiplication and division. For 3x + 5 = 20, subtract 5 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5. Always check your answer by putting it back into the original equation.
The quadratic formula
A quadratic equation has the form ax² + bx + c = 0. The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, b² − 4ac, is called the discriminant. If it is positive there are two answers, if it is zero there is one answer, and if it is negative there is no real answer. For x² − 5x + 6 = 0, a = 1, b = −5 and c = 6, so x = (5 ± √(25 − 24)) ÷ 2 = 3 or 2.
Systems of equations
A system of equations is two equations with the same two unknowns, such as x + y = 5 and x − y = 1. The solution is the pair (x, y) that satisfies both equations at once — the point where the two lines cross. You can solve a system by substitution, by elimination (adding or subtracting the equations) or with the determinant formula D = a1b2 − a2b1. If D = 0 there is no unique solution: the lines are parallel or the same line.
Worked examples
Example 1: Solve 3x + 5 = 20
Subtract 5 from both sides: 3x = 15. Divide both sides by 3: x = 5. Check: 3 × 5 + 5 = 20. Correct.
Example 2: Solve x² − 5x + 6 = 0
Using the quadratic formula with a = 1, b = −5 and c = 6: x = (5 ± √(25 − 24)) ÷ 2 = (5 ± 1) ÷ 2, so x = 3 or x = 2. Check: 3² − 15 + 6 = 0 and 2² − 10 + 6 = 0. Correct.
Example 3: Solve x + y = 5 and x − y = 1
Add the equations: 2x = 6, so x = 3. Put x = 3 into x + y = 5: 3 + y = 5, so y = 2. Check: 3 − 2 = 1. Correct.
Practice questions
- Solve 2x + 7 = 15.
- Solve x² − 9 = 0.
- Solve 4x − 3 = 2x + 9.
- Expand (x + 2)(x + 3).
- Solve x + 2y = 7 and 3x − y = 7.
Frequently Asked Questions
What is algebra used for?
Algebra is used to solve problems with unknown numbers, from working out a phone bill to designing bridges and programming computers.
What is the difference between an expression and an equation?
An expression, such as 2x + 3, has no equals sign and cannot be solved. An equation, such as 2x + 3 = 11, has an equals sign and can be solved for x.
How do I check my answer?
Put your answer back into the original equation. If both sides are equal, your answer is correct.
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