Number & Algebra

Algebraic Processes

JSS 1 · JSS 2 · JSS 3

✓By the end of this lesson students will be able to simplify algebraic expressions by collecting like terms.
✓By the end of this lesson students will be able to solve simple linear equations involving one unknown.
✓By the end of this lesson students will be able to generate tables of values for simple linear equations.
✓By the end of this lesson students will be able to plot points and draw graphs of simple linear equations on a Cartesian plane.

Key concepts

Algebraic Expressions and Simplification

An algebraic expression is a combination of numbers, variables (letters representing unknown values), and mathematical operations (+, -, ×, ÷). A 'term' in an expression is a single number, a single variable, or a product of numbers and variables. 'Like terms' are terms that have the same variables raised to the same power. For example, 3x and 5x are like terms, but 3x and 5y are not. To simplify an algebraic expression, we combine (add or subtract) the like terms.

Linear Equations

An equation is a mathematical statement that shows two expressions are equal. A 'linear equation' is an equation where the highest power of the variable (the unknown) is 1. For example, x + 5 = 10 is a linear equation. To 'solve' a linear equation means to find the value of the unknown variable that makes the equation true. We solve equations by performing the same operation (addition, subtraction, multiplication, or division) on both sides of the equality sign to isolate the unknown variable. Another common method is 'transposition', where a term is moved from one side of the equation to the other, changing its sign (e.g., + becomes -, × becomes ÷).

ax + b = c (where a, b, c are numbers and x is the unknown)

Simple Equations and Graphs

The graph of a linear equation is always a straight line. To draw the graph of a simple linear equation (e.g., y = x + 2), we first create a 'table of values'. This involves choosing several values for 'x' (usually small integers like -2, -1, 0, 1, 2) and substituting them into the equation to find the corresponding values for 'y'. Each pair of (x, y) values forms a 'coordinate' or 'ordered pair'. These coordinates are then plotted on a 'Cartesian plane' (a grid with a horizontal x-axis and a vertical y-axis). Once the points are plotted, a straight line is drawn through them to represent the graph of the equation.

y = mx + c (where m and c are constants)

Key facts to remember

1Variables are letters used to represent unknown numbers.
2Like terms have the same variables raised to the same power and can be added or subtracted.
3An equation states that two expressions are equal.
4To solve an equation, perform the same operation on both sides to isolate the unknown.
5Transposition involves moving a term to the other side of the equation, changing its operation (e.g., + becomes -, × becomes ÷).
6The graph of a linear equation is always a straight line.
7A table of values helps to find coordinates (x, y) for plotting a graph.
8The Cartesian plane uses an x-axis (horizontal) and a y-axis (vertical) to plot points.

Worked examples

Example 1

Simplify the expression: 7a + 3b - 2a + 5b.

IIdentify like terms: (7a and -2a) and (3b and 5b).

IIGroup the like terms together: 7a - 2a + 3b + 5b

IIICombine the 'a' terms: (7 - 2)a = 5a

IVCombine the 'b' terms: (3 + 5)b = 8b

VWrite the simplified expression.

Answer

5a + 8b

Remember to include the sign in front of each term when grouping.

Example 2

Solve the equation: 3x - 7 = 8.

IAdd 7 to both sides of the equation (or transpose -7 to the right side, changing it to +7):

II3x - 7 + 7 = 8 + 7

III3x = 15

IVDivide both sides by 3 (or transpose 3, changing it from multiplication to division):

V3x / 3 = 15 / 3

VIx = 5

Answer

x = 5

Always perform the same operation on both sides to maintain balance.

Example 3

Draw the graph of the equation y = x + 1 for x values from -2 to 2.

ICreate a table of values:

IIWhen x = -2, y = -2 + 1 = -1. Point: (-2, -1)

IIIWhen x = -1, y = -1 + 1 = 0. Point: (-1, 0)

IVWhen x = 0, y = 0 + 1 = 1. Point: (0, 1)

VWhen x = 1, y = 1 + 1 = 2. Point: (1, 2)

VIWhen x = 2, y = 2 + 1 = 3. Point: (2, 3)

VIIDraw a Cartesian plane with x-axis and y-axis, labelling the origin (0,0) and scales.

VIIIPlot the points: (-2, -1), (-1, 0), (0, 1), (1, 2), (2, 3).

9Draw a straight line passing through all the plotted points.

10Label the line with its equation: y = x + 1.

Answer

A straight line graph passing through the points (-2, -1), (-1, 0), (0, 1), (1, 2), (2, 3).

Use a ruler to draw straight lines and ensure your axes are clearly labelled with scales.

Common mistakes

✗Adding or subtracting unlike terms (e.g., trying to combine 3x and 2y).
✗Forgetting to change the sign of a term when transposing it across the equality sign.
✗Making arithmetic errors when substituting values into an equation for a table of values.
✗Not drawing a straight line when graphing a linear equation, or not using a ruler.
✗Confusing the x-coordinate and y-coordinate when plotting points (e.g., plotting (y, x) instead of (x, y)).

Exam tips

★Always show all your working steps clearly, especially for solving equations, as marks are awarded for method.
★Double-check your arithmetic, especially when dealing with negative numbers.
★When drawing graphs, use a sharp pencil and a ruler. Label your axes and the line clearly.
★Verify your solution to an equation by substituting your answer back into the original equation to see if it holds true.

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