Algebra

The One Thing

Algebraic manipulation depends on applying the distributive law, standard identities, and the laws of equality accurately. Expansion and factorisation are inverse processes, while equations are solved by preserving equality and checking that the resulting values satisfy the original conditions.

Definitions and Results

• Algebraic expression: A combination of numbers, variables, and operations, such as $3x^2+5x-2$.
• Term: A part of an expression separated by plus or minus signs; for example, $4x$ and $-7$ are terms.
• Coefficient: The numerical factor multiplying a variable; in $6y$, $6$ is the coefficient of $y$.
• Constant: A number without a variable, such as $9$ in $2x+9$.
• Like terms: Terms having the same variables raised to the same powers, such as $3x$ and $-5x$. Only like terms may be combined: $3x+5x=8x$, whereas $3x+5y$ cannot be simplified further.
• Distributive law: Multiplication over a bracket:

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

• Expansion: Removing brackets and writing a product as a sum or difference of terms.
• Algebraic identity: An equality that is true for all permitted values of its variables. The principal identities are:

$$(a+b{)}^2=a^2+2ab+b^2,$$ $$(a-b{)}^2=a^2-2ab+b^2,$$ $$(a+b)(a-b)=a^2-b^2.$$ Further useful identities are: $$(x+a)(x+b)=x^2+(a+b)x+ab,$$ $$(x-a)(x-b)=x^2-(a+b)x+ab.$$

• Factorisation: Writing an expression as a product of two or more factors.
• Common factor: A factor shared by every term of an expression and taken outside brackets. The highest common factor should be taken out whenever one exists.
• Linear equation: An equation in which the highest power of the variable is $1$, such as $3x-4=11$.
• Simultaneous equations: Two or more equations solved together to find values satisfying all the equations.
• Substitution method: Finding one variable in terms of another and replacing it in the second equation.
• Elimination method: Adding or subtracting equations, sometimes after multiplying them, to remove one variable.
• Solution or ordered pair: The values of the variables that satisfy an equation or a pair of simultaneous equations, written as $(x,y)$ when appropriate.
• Equation-solving condition: Any operation performed on one side of an equation must also be performed on the other side.
• Simultaneous-equation outcomes: A pair of linear simultaneous equations may have one solution, no solution, or infinitely many solutions. In typical school problems, the objective is usually to find the unique ordered pair.

Worked Methods

Expansion using the distributive law

• Multiply each term outside a bracket by every term inside it.
• Preserve the signs of all terms.
• Collect like terms if necessary.

For a single bracket: $$a(b+c)=ab+ac,a(b-c)=ab-ac.$$

For two binomials: $$(a+b)(c+d)=ac+ad+bc+bd.$$

Every term inside a bracket must be multiplied, including terms with negative signs.

Expansion using identities

For a square:

• Square the first term.
• Add or subtract twice the product of the two terms, using the correct sign.
• Square the second term.

Thus: $$(a+b{)}^2=a^2+2ab+b^2,$$ and $$(a-b{)}^2=a^2-2ab+b^2.$$

For a difference of squares: $$(a+b)(a-b)=a^2-b^2.$$

The identities are algebraic identities, so they are valid for all permitted values of the variables.

Factorisation by taking out a common factor

• Identify the highest common factor of every term.
• Divide each term by that factor.
• Write the common factor outside brackets.
• Check by re-expanding.

Factorisation expresses an expression as a product, reversing the process of expansion.

Factorisation using a difference of squares

For an expression of the form $a^2-b^2$:

• Recognise the two square terms.
• Use the identity

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

Factorisation of a quadratic with positive middle-term coefficient

For an expression of the form $$x^2+(p+q)x+pq:$$

• Identify two numbers $p$ and $q$ whose sum is the coefficient of $x$.
• Confirm that their product is the constant term.
• Write

$$x^2+(p+q)x+pq=(x+p)(x+q).$$

Factorisation of a quadratic with negative middle-term coefficient

For an expression of the form $$x^2-(p+q)x+pq:$$

• Identify $p$ and $q$ whose sum is the magnitude of the coefficient of $x$.
• Confirm that their product is the constant term.
• Write

$$x^2-(p+q)x+pq=(x-p)(x-q).$$

Solving a linear equation

• Simplify both sides by expanding brackets and combining like terms where necessary.
• Perform the same operation on both sides of the equality sign.
• Isolate the variable.
• Check the result by substituting it into the original equation.

For example, the equation $$3x-4=11$$ is solved by adding $4$ to both sides and then dividing by $3$. The resulting value must be checked in the original equation.

When removing brackets in an equation, multiply every term inside the bracket, including terms with negative signs.

For equations involving fractions:

• Identify the lowest common denominator.
• Multiply every term on both sides by that lowest common denominator.
• Solve the resulting equation.
• Check the answer in the original fractional equation.

Solving simultaneous equations by substitution

• Rearrange one equation to express one variable in terms of the other.
• Substitute that expression into the other original equation.
• Solve the resulting equation for one variable.
• Substitute the value back into the rearranged equation.
• State the solution as an ordered pair $(x,y)$.
• Check both values in both original equations.

Solving simultaneous equations by elimination

• Write the equations in a consistent order for the variables.
• Make the coefficients of one variable equal or opposites, multiplying one or both equations if necessary.
• Add or subtract the equations to eliminate that variable.
• Solve for the remaining variable.
• Substitute the value into one original equation to find the other variable.
• State the ordered pair and check it in both original equations.

Translating word problems into equations

• Define the variables clearly.
• Identify the relationships described in the problem.
• Translate those relationships into equations.
• Solve the resulting linear or simultaneous equations.
• Interpret the solution in the context of the problem and check it against the original relationships.

Where It Goes Wrong

• Combining unlike terms, such as treating $3x+5y$ as simplifiable; only terms with the same variables raised to the same powers are like terms.
• Applying the distributive law to only the first term in a bracket; every term inside the bracket must be multiplied, including terms with negative signs.
• Using the wrong sign in a square identity; $(a-b{)}^2$ contains $-2ab$, whereas $(a+b{)}^2$ contains $+2ab$.
• Forgetting that factorisation should begin with the highest common factor whenever one exists.
• Performing an operation on only one side of an equation, rather than applying the same operation to both sides.
• Failing to check a solution in the original equation or in every equation of a simultaneous system.

What Gets Asked

• Simplify algebraic expressions by identifying terms, coefficients, constants, and like terms.
• Expand single brackets and two binomials using the distributive law.
• Expand squares using $(a+b{)}^2$ and $(a-b{)}^2$, including correct signs.
• Expand and use the difference-of-squares identity $(a+b)(a-b)=a^2-b^2$.
• Factorise expressions by taking out the highest common factor.
• Factorise differences of squares.
• Factorise quadratics of the forms $x^2+(p+q)x+pq$ and $x^2-(p+q)x+pq$.
• Solve linear equations, including equations involving brackets and fractions.
• Solve simultaneous equations using substitution.
• Solve simultaneous equations using elimination.
• Translate word problems into equations by defining variables and identifying relationships.
• Verify a proposed value or ordered pair by substitution into the original equation or equations.