Algebra

The One Thing

Algebra represents relationships between quantities using expressions, equations, inequalities, functions and models. Correct manipulation preserves mathematical relationships, while interpretation requires attention to domains, units, assumptions and the context of the result.

Definitions and Results

• Variable: A letter or symbol representing a number or quantity that may change.
• Algebraic expression: A mathematical phrase made from numbers, variables and operations, such as $3x+5$.
• Equation: A statement that two expressions are equal, using an equals sign.
• Inequality: A statement comparing quantities using $<$, $>$, $\le$ or $\ge$.
• Function: A relation in which every input has exactly one output.
• Domain: The set of possible input values for a function.
• Range: The set of output values produced by a function.
• Independent variable: The input variable, commonly $x$, whose value is selected or changes.
• Dependent variable: The output variable, commonly $y$, whose value depends on the input.
• Linear function: A function with a constant rate of change whose graph is a straight line.
• Gradient: The rate of change of a straight line:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• $y$-intercept: The point where a graph crosses the $y$-axis. In $y=mx+b$, it is $b$.
• Quadratic function: A function of the form

$$y=a{x}^2+bx+c,a\ne 0,$$ whose graph is a parabola.

• Factorisation: Rewriting an expression as a product of factors, for example:

$$x^2+5x+6=(x+2)(x+3).$$

• Solution: A value or set of values that makes an equation or inequality true.
• Simultaneous equations: Two or more equations solved together to find values satisfying all equations.
• Algebraic model: An equation, function or inequality representing a real-world relationship.
• Exponential growth or decay: A pattern in which a quantity is repeatedly multiplied by a constant factor, commonly written:

$$y=a{b}^x.$$

• Distributive law:

$$a(b+c)=ab+ac.$$

• Collecting like terms: Terms with the same variable and power can be combined, for example:

$$3x+2x=5x.$$

• Index laws:

$$a^m\times a^n=a^{m+n},$$ $$a^m÷a^n=a^{m-n},a\ne 0,$$ $$(a^m{)}^n=a^{mn}.$$

• Binomial identity:

$$(a+b{)}^2=a^2+2ab+b^2.$$

• Linear equation principle: The same operation must be performed on both sides of an equation.
• Inequality reversal: Multiplying or dividing an inequality by a negative number reverses its sign.
• Linear equation form:

$$y=mx+b,$$ where $m$ is the gradient and $b$ is the $y$-intercept.

• Distance between two points:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

• Midpoint of two points:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

• Quadratic formula: For $a{x}^2+bx+c=0$,

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Axis of symmetry of a quadratic:

$$x=-\frac{b}{2a}.$$

• Discriminant: The expression $b^2-4ac$. A positive discriminant gives two real solutions, zero gives one repeated real solution, and a negative discriminant gives no real solutions.
• Function representations: A function may be represented by a rule, table of values, mapping diagram or graph.
• Vertical-line test: A graph represents a function if every vertical line intersects it at most once.
• Simultaneous-equation solution: The solution is the intersection point of the graphs.
• Algebraic modelling conditions: A model requires defined variables, stated assumptions, an appropriate domain, suitable units and interpretation of the result.

Worked Methods

Simplifying and expanding algebraic expressions

• Apply the distributive law:

$$a(b+c)=ab+ac.$$

• Expand each term.
• Collect like terms, combining only terms with the same variable and power.
• Use the index laws where powers are involved:

$$a^m\times a^n=a^{m+n},a^m÷a^n=a^{m-n},(a^m{)}^n=a^{mn}.$$

• For a squared binomial, use:

$$(a+b{)}^2=a^2+2ab+b^2.$$

For example, like terms combine as: $$3x+2x=5x.$$

Solving a linear equation

• Simplify both sides, including expansion and collection of like terms.
• Perform the same operation on both sides to preserve equality.
• Isolate the variable.
• Check the result by substitution into the original equation.

Equivalent algebraic transformations preserve the solution set when applied correctly to both sides.

Solving an inequality

• Simplify and collect terms.
• Perform equivalent operations on both sides.
• Isolate the variable.
• If multiplying or dividing by a negative number, reverse the inequality sign.
• Express the result using interval notation, a number line or a shaded region where appropriate.

Unlike an equation, an inequality generally describes a range of possible values rather than one answer.

Working with linear functions and graphs

• Express the function in the form:

$$y=mx+b.$$

• Identify $m$ as the gradient and $b$ as the $y$-intercept.
• To calculate the gradient between two points, use:

$$m=\frac{y_2-y_1}{x_2-x_1}.$$

• Use the gradient to determine the rate of change.
• Interpret intercepts and other graph features in context.

Graphs provide information about solutions, intercepts and rates of change.

Calculating distance and midpoint

For points $(x_1,y_1)$ and $(x_2,y_2)$:

• Calculate the horizontal and vertical differences.
• Substitute into the distance formula:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}.$$

• For the midpoint, average the two $x$-coordinates and the two $y$-coordinates:

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).$$

Solving a quadratic equation

For an equation in the form: $$a{x}^2+bx+c=0:$$

• Attempt to factorise the expression.
• Alternatively, complete the square, use a graph or apply the quadratic formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• If using the discriminant $b^2-4ac$, determine the number of real solutions:

- positive: two real solutions; - zero: one repeated real solution; - negative: no real solutions.

• For $y=a{x}^2+bx+c$, find the axis of symmetry using:

$$x=-\frac{b}{2a}.$$

• Check the solutions by substitution or graphing.

A stated example of factorisation is: $$x^2+5x+6=(x+2)(x+3).$$

Solving simultaneous linear equations

• Write the equations clearly.
• Solve by substitution, elimination or graphing.
• If using substitution, rearrange one equation and substitute into the other.
• If using elimination, add or subtract equations to remove one variable.
• If graphing, identify the intersection point.
• Check the resulting values in both original equations.

The solution is the intersection point of the graphs and must satisfy all equations.

Representing and testing a function

• Identify the input and output variables.
• State the function rule, or construct a table of values, mapping diagram or graph.
• Identify the domain and range.
• Test whether each input has exactly one output.
• For a graph, apply the vertical-line test: every vertical line must intersect the graph at most once.

Constructing and analysing an algebraic model

• Define the variables.
• Identify the assumptions.
• Form an equation, function or inequality representing the situation.
• Apply any necessary domain restrictions, such as non-negative time or a whole-number number of people.
• Solve or analyse the model.
• Include appropriate units and rounding.
• Interpret the result in context.
• Check the result by substitution, estimation, graphing or considering whether it makes sense in the situation.

Graphing software or spreadsheets may assist investigation, but the mathematical solution must still be interpreted and checked.

Where It Goes Wrong

• Applying the distributive law incorrectly, or failing to collect only terms with the same variable and power.
• Using index laws without observing their conditions, particularly $a^m÷a^n=a^{m-n}$, which requires $a\ne 0$.
• Performing an operation on only one side of an equation rather than applying it to both sides.
• Failing to reverse an inequality sign when multiplying or dividing by a negative number.
• Treating an inequality as though it normally has one solution, instead of representing a range using interval notation, a number line or a shaded region.
• Ignoring domain restrictions, units, rounding, assumptions or the contextual meaning of a model; for example, time may need to be non-negative and the number of people may need to be a whole number.

What Gets Asked

• Simplify algebraic expressions using the distributive law, collection of like terms and index laws.
• Expand expressions using $(a+b{)}^2=a^2+2ab+b^2$.
• Factorise expressions such as:

$$x^2+5x+6=(x+2)(x+3).$$

• Solve linear equations and verify solutions by substitution.
• Solve inequalities, including cases requiring reversal of the inequality sign.
• Represent functions using rules, tables, mapping diagrams and graphs, and identify domain and range.
• Determine whether a graph represents a function using the vertical-line test.
• Find the gradient and $y$-intercept of a linear function in the form $y=mx+b$.
• Calculate the distance and midpoint between two points.
• Solve quadratic equations by factorising, completing the square, graphing or using:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Use the discriminant to determine the number of real solutions and find the axis of symmetry:

$$x=-\frac{b}{2a}.$$

• Solve simultaneous linear equations by substitution, elimination or graphing.
• Form and interpret algebraic models involving growth, motion, costs, measurement and exponential growth or decay:

$$y=a{b}^x.$$

• State assumptions, impose realistic domain restrictions, include units and rounding, and assess whether a model’s prediction is reasonable.
• Interpret graphs in terms of solutions, intercepts, rates of change, turning points and intersections.