Numbers & Algebra
Algebraic Expressions, Equations and Graphs
Grade 7 · Grade 8 · Grade 9
- ✓Define and identify components of algebraic expressions, including terms, variables, coefficients, and constants.
- ✓Simplify algebraic expressions by collecting like terms.
- ✓Solve linear equations in one variable using inverse operations.
- ✓Understand the Cartesian plane and plot points using ordered pairs.
- ✓Draw graphs of linear equations by plotting points from a table of values.
Key concepts
A mathematical phrase that contains numbers, variables (letters), and operation signs (+, -, ×, ÷), but does not contain an equals sign.
Parts of an algebraic expression that are separated by addition (+) or subtraction (-) signs.
A letter (e.g., x, y, a) used to represent an unknown numerical value that can change.
The numerical factor that multiplies a variable in a term. If no number is written, the coefficient is 1 (or -1 if there's a minus sign).
A term in an algebraic expression that does not contain a variable; its value remains fixed.
Terms that have the exact same variables raised to the exact same powers. Only like terms can be added or subtracted.
The process of combining like terms in an expression to make it shorter and easier to work with.
A mathematical statement that shows two expressions are equal, where the highest power of the variable is 1. It contains an equals sign.
The process of finding the value(s) of the variable(s) that make the equation true. This is done by isolating the variable using inverse operations on both sides of the equation.
A two-dimensional plane formed by two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), intersecting at the origin (0; 0).
A pair of numbers (x; y) that specifies the exact location of a point on the Cartesian plane. The first number is the x-coordinate, and the second is the y-coordinate.
A straight line drawn on the Cartesian plane that represents all the possible solutions (x; y) to a linear equation in two variables.
Key facts to remember
- 1An algebraic expression does not have an equals sign, while an equation does.
- 2Only like terms can be added or subtracted in an algebraic expression.
- 3To solve an equation, perform the same operation on both sides to maintain balance and isolate the variable.
- 4The Cartesian plane uses ordered pairs (x; y) to locate points, where x is the horizontal coordinate and y is the vertical coordinate.
- 5The graph of a linear equation is always a straight line.
- 6The coefficient includes the sign directly in front of the number (e.g., in -3x, the coefficient is -3).
- 7The origin of the Cartesian plane is the point (0; 0).
Worked examples
Example 1
Simplify the following algebraic expression: 7x + 3y - 2x - 5y + 8
Answer
5x - 2y + 8
Remember to include the sign in front of each term when grouping.
Example 2
Solve for x: 4x - 7 = 13
Answer
x = 5
You can check your answer by substituting x=5 back into the original equation: 4(5) - 7 = 20 - 7 = 13. Since 13 = 13, the solution is correct.
Example 3
Draw the graph of the linear equation y = -x + 2 for x values from -2 to 2.
Answer
The graph will be a straight line passing through the points (-2; 4), (-1; 3), (0; 2), (1; 1), and (2; 0). It will slope downwards from left to right.
Always use a ruler to draw straight lines and ensure your axes are clearly labelled with appropriate scales.
Common mistakes
- ✗Adding or subtracting terms that are not like terms (e.g., 3x + 2y ≠ 5xy).
- ✗Forgetting to perform an operation on *both* sides of an equation, leading to an incorrect solution.
- ✗Incorrectly handling negative signs when combining terms or applying inverse operations.
- ✗Confusing the x-coordinate and y-coordinate when plotting points (e.g., plotting (3; 2) instead of (2; 3)).
- ✗Not extending the line with arrows and labelling the graph of a linear equation.
Exam tips
- ★Always show all your working steps clearly and logically, especially when simplifying expressions or solving equations.
- ★Check your solutions to equations by substituting your answer back into the original equation to verify its correctness.
- ★When drawing graphs, use a sharp pencil and a ruler. Label your axes (x and y), the origin (0), and the equation of the line.
- ★Pay close attention to positive and negative signs throughout your calculations; a single sign error can lead to an incorrect answer.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.