Algebra: Solving Equations & Inequalities
Algebra uses letters and symbols to represent variables, formulate equations, and solve mathematical problems systematically. For GCSE Maths, fluency with expanding brackets, factorising expressions, rearranging formulae, and solving linear, quadratic, and simultaneous equations underpins both Foundation and Higher papers.
- Variable
- A letter used to represent an unknown number.
- Equation
- A mathematical statement showing that two expressions are equal.
- Inequality
- A relationship between two expressions that are not strictly equal, such as "greater than" or "less than".
Quadratic Parabola Plotter, Roots, Turning Point & Quadratic Formula
Adjust coefficients a, b, and c in y = ax² + bx + c to see how the parabola, roots (x-intercepts), and turning point move on the graph.
y = x² - 2x - 3
(-2)² − 4(1)(-3) = 16 → 2 real roots (crosses x-axis twice)x = 1.00b in brackets when squaring: (-2)² = 4 (always positive!). Typing -2² into a calculator without brackets gives -4 and loses the accuracy mark.Key Concepts & Exam Definitions (5)
- Variable
- A letter used to represent an unknown number.
- Equation
- A mathematical statement showing that two expressions are equal.
- Inequality
- A relationship between two expressions that are not strictly equal, such as "greater than" or "less than".
- Simultaneous Equations
- Two or more algebraic equations that share variables and are true at the same time.
- Substitution
- Replacing variables in an expression or equation with numbers or other expressions.
Quick-Fire Flashcards (5)
Click any card to check the answer, then mark it as Known to track your progress.
What does the symbol ≤ mean?
Solve for x: 3x - 5 = 10
What happens to the inequality sign when you multiply or divide by a negative number?
Solve the inequality: -2x > 8
In the equation y = mx + c, what does m represent?
Exam-Style Questions & Mark Schemes (3)
Write or think through your answer first, then click Check Mark Scheme to see the exact points examiners award marks for.
- Q1
Solve the equation: 4(x + 2) = 24
Click “Check Mark Scheme” to compare your answer, x = 4. Divide both sides by 4 to get x + 2 = 6, then subtract 2.
- Q2
Solve the simultaneous equations: x + y = 10 and x - y = 4
Click “Check Mark Scheme” to compare your answer, x = 7, y = 3. Adding the two equations gives 2x = 14, so x = 7. Substitute x back to get y = 3.
- Q3
List all the integer values of n that satisfy the inequality: -2 < n ≤ 3
Click “Check Mark Scheme” to compare your answer, -1, 0, 1, 2, 3
Common Exam Questions (3)
What is the golden rule of algebra?
An equation must remain balanced: any operation (addition, subtraction, multiplication, or division) performed on the left-hand side of the equals sign must also be performed identically on the right-hand side.
How do you solve simultaneous equations?
Simultaneous equations can be solved algebraically by elimination (multiplying equations so one variable has matching coefficients and adding or subtracting) or by substitution (rearranging one equation for x or y and substituting into the other), or graphically by finding the coordinates where the two lines intersect.
What is the difference between an equation and an inequality?
An equation states that two expressions are equal (=) and is satisfied by specific values of the variable, whereas an inequality compares the relative size of two expressions using <, >, ≤, or ≥ and is satisfied by a continuous range of values.
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