Algebra

The One Thing

GCSE algebra represents patterns and relationships symbolically, then uses valid operations to simplify expressions, solve equations and inequalities, describe sequences, and interpret graphs. Accuracy depends on identifying the structure of the problem, preserving equality or inequality meaning, and checking the result in context.

Definitions and Results

• Variable: A letter or symbol representing an unknown or changing value.
• Constant: A fixed number that does not change.
• Coefficient: The number multiplying a variable, such as $5$ in $5x$.
• Term: A number, variable, or product of numbers and variables separated by plus or minus signs.
• Expression: A mathematical statement containing numbers, variables, and operations but no equals sign.
• Equation: A statement that two expressions are equal, shown using an equals sign.
• Inequality: A comparison showing that one value is greater than, less than, or possibly equal to another.
• Like terms: Terms with the same variable parts and powers, such as $3x$ and $7x$.
• Expanding: Removing brackets by multiplying every term inside a bracket by the factor outside it.
• Factorising: Writing an expression as a product of factors, often by taking out a common factor or reversing expansion.
• Identity: An equation true for every permitted value of the variable, such as

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

• Linear equation: An equation in which the variable has highest power $1$, such as

$$3x+4=19.$$

• Quadratic equation: An equation involving a squared variable, usually written as

$$a{x}^2+bx+c=0.$$

• Changing the subject: Rearranging a formula so that a chosen variable is alone on one side.
• Sequence: An ordered list of numbers following a rule or pattern.
• Term-to-term rule: A rule explaining how to obtain one sequence term from the previous term.
• Position-to-term rule: A formula giving the value of a sequence term from its position number.
• Arithmetic sequence: A sequence with a constant common difference between consecutive terms.
• Gradient: The steepness of a line, calculated as change in $y$ divided by change in $x$.
• $y$-intercept: The point where a graph crosses the $y$-axis.
• Function: A rule mapping each input value to exactly one output value.
• Reciprocal graph: A graph such as

$$y=\frac{1}{x},$$ with two separate curved branches and asymptotes.

• Order of operations: Use brackets, powers, division and multiplication, then addition and subtraction.
• Collecting like terms:

$$3x+5x=8x,$$ but $3x$ and $5x^2$ cannot be combined.

• Expanding:

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

• Common-factor factorising:

$$6x+9=3(2x+3).$$

• Difference of two squares:

$$a^2-b^2=(a-b)(a+b).$$

• Solving linear equations: Apply inverse operations in a balanced way to both sides.
• Inequalities: Multiplying or dividing by a negative number reverses the inequality sign. Thus,

$$-2x>6\Rightarrow x<-3.$$

• Zero-product rule: If

$$pq=0,$$ then $p=0$ or $q=0$.

• Quadratic formula: For

$$a{x}^2+bx+c=0,$$ $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}.$$

• Discriminant: The value $b^2-4ac$ determines the number of real quadratic solutions:

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

• Arithmetic-sequence formula: If the first term is $a$ and the common difference is $d$, the $n$th term is

$$a+(n-1)d.$$

• Straight-line equation:

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

• Gradient between two points:

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

• Parallel lines: Parallel straight lines have equal gradients.
• Perpendicular lines: Perpendicular non-vertical lines have gradients whose product is $-1$.
• Standard graphs: $y=x^2$ is a parabola. The graph

$$y=\frac{1}{x}$$ has asymptotes $x=0$ and $y=0$.

• Simultaneous equations: These may be solved by substitution, elimination, or graphical intersection.
• Graphical solutions: The solutions of an equation are represented by the $x$-coordinates where a graph crosses or meets the $x$-axis.
• Exact values: Use exact values, such as fractions or surds, where appropriate, and check numerical answers by substitution.
• Algebraic modelling: Define variables clearly, form an equation or inequality from the situation, solve it, and interpret the answer in context.

Worked Methods

Simplifying expressions

• Apply the order of operations: brackets, powers, division and multiplication, then addition and subtraction.
• Identify terms with the same variable parts and powers.
• Collect only like terms.
• For example:

$$3x+5x=8x.$$

• Do not combine unlike terms:

$$3x\text{ and }5x^2$$ cannot be collected.

Expanding brackets

• Multiply the factor outside the bracket by every term inside it.
• Use

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

• Simplify the resulting expression by collecting like terms if necessary.

Factorising

• Identify a common factor in every term.
• Take out the greatest suitable common factor.
• Write the remaining factors in brackets.
• For example:

$$6x+9=3(2x+3).$$

• For a difference of two squares, use

$$a^2-b^2=(a-b)(a+b).$$

Solving a linear equation

• Apply inverse operations to isolate the variable.
• Perform the same valid operation on both sides to preserve equality.
• Continue until the variable is alone.
• Check the result by substituting it into the original equation.
• A linear equation has the variable to the highest power of $1$, as in

$$3x+4=19.$$

Solving a linear inequality

• Apply the same valid operation to both sides.
• Isolate the variable.
• If multiplying or dividing by a negative number, reverse the inequality sign.
• For example:

$$-2x>6.$$ Dividing by $-2$ gives $$x<-3.$$

• Represent the solution as a range of values, using inequality notation or a number line.

Solving a quadratic by factorising

• Rearrange the equation into

$$a{x}^2+bx+c=0.$$

• Factorise where suitable.
• Apply the zero-product rule: if $pq=0$, then $p=0$ or $q=0$.
• Solve each resulting linear equation.
• Check the solutions in the original equation.

Solving a quadratic using the quadratic formula

• Write the equation in the form

$$a{x}^2+bx+c=0.$$

• Identify $a$, $b$, and $c$.
• Substitute them into

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

• Evaluate both the positive and negative square-root cases.
• Use the discriminant $b^2-4ac$ to determine whether there are two, one repeated, or no real solutions.

Changing the subject of a formula

• Identify the variable that must be isolated.
• Undo operations in reverse order using inverse operations.
• Apply each operation consistently to preserve equality.
• Continue until the chosen variable is alone on one side.
• Check the rearrangement by substituting suitable values.

Finding terms in a sequence

• Examine consecutive terms to identify a pattern.
• State the term-to-term rule if the relationship between adjacent terms is required.
• For an arithmetic sequence, identify the first term $a$ and common difference $d$.
• Use the position-to-term rule

$$a+(n-1)d$$ to find a distant term.

• Check the formula against the first few terms.

Working with straight-line graphs

• Compare the equation with

$$y=mx+c.$$

• Identify $m$ as the gradient and $c$ as the $y$-intercept.
• For two points $(x_1,y_1)$ and $(x_2,y_2)$, calculate

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

• Use equal gradients to identify parallel lines.
• For perpendicular non-vertical lines, check that the product of the gradients is $-1$.

Interpreting standard graphs

• Recognise $y=x^2$ as a parabola.
• Recognise

$$y=\frac{1}{x}$$ as a reciprocal graph with two separate curved branches.

• Identify the reciprocal graph’s asymptotes:

$$x=0\text{and}y=0.$$

• Interpret roots as the $x$-coordinates where a graph crosses or meets the $x$-axis.

Solving simultaneous equations

• Choose substitution, elimination, or graphical intersection.
• Use substitution by expressing one variable in terms of the other and replacing it in the second equation.
• Use elimination by combining equations to remove one variable.
• Use graphical intersection by finding the coordinates where the graphs meet.
• Substitute the values back into the original equations to check them.

Algebraic modelling

• Define the variables clearly.
• Translate the situation into an equation or inequality.
• Solve the equation or inequality using appropriate algebraic methods.
• Interpret the answer in the original context.
• Check algebraic accuracy, sensible values, restrictions, and the requirements of the problem.

Where It Goes Wrong

• Combining unlike terms, such as treating $3x$ and $5x^2$ as like terms.
• Failing to multiply every term inside a bracket when expanding:

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

• Applying an operation to only one side of an equation rather than maintaining balance on both sides.
• Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
• Omitting one of the two solutions from the $\pm$ in the quadratic formula or misidentifying $a$, $b$, and $c$.
• Giving an algebraic result without checking it by substitution, considering restrictions, or interpreting it in context.

What Gets Asked

This material supports questions requiring students to:

• simplify expressions using order of operations and collect like terms;
• expand brackets and factorise expressions, including differences of two squares;
• distinguish variables, constants, coefficients, terms, expressions, equations, inequalities, and identities;
• solve linear equations and inequalities, including inequalities involving multiplication or division by negative numbers;
• solve quadratic equations by factorising, the zero-product rule, or the quadratic formula;
• use the discriminant to determine the number of real solutions;
• change the subject of a formula;
• find term-to-term and position-to-term rules for sequences;
• determine terms in arithmetic sequences using $a+(n-1)d$;
• calculate gradients and $y$-intercepts from equations or coordinates;
• identify parallel and perpendicular lines;
• interpret $y=x^2$ and $y=1/x$, including the reciprocal graph’s asymptotes;
• solve simultaneous equations by substitution, elimination, or graphical intersection;
• identify equation solutions from graph intersections with the $x$-axis;
• construct and solve algebraic models from contextual situations;
• give exact values using fractions or surds and verify answers by substitution.