Understanding algebraic linear equations is essential for securing top grades in your Maths exams, as it forms the foundation of many questions that will appear on your papers. This guide is designed to empower you with the fundamental knowledge and techniques necessary for excelling in your exams, providing example questions and explaining key concepts to help you approach algebraic linear equations with confidence.

 

Basic Algebraic Expressions

An algebraic expression contains variables, constants, and the four operations (Do you recall BODMAS?). 

Algebraic BODMAS

 

Examples of basic expressions include:

  1. Linear Expression: 3x+5
  2. Quadratic Expression: 2×2-4x+7
  3. Exponential Expression: 2x+1

 

Like & Unlike Terms in Algebraic Expressions

Like terms are terms that share identical variables. For example, 4x, -9x and 12x share the same variable of x, which can be added to and subtracted from each other.

 

For instance, consider the expression: 21xy +4xy=25xy

However, unlike terms in algebra involve different variables. For example, 7xy, -8y and 2x, have two different variables x and y, thus they cannot be added and subtracted freely within the algebraic equation.

For instance, consider the expression: 19x+3y

We cannot simplify this expression any further as x and y are unknown.

 

Algebraic Expressions as Word Statements

Understanding how to interpret and formulate algebraic expressions as word statements is essential for tackling questions with clarity and precision. Below, we have compiled a list of the most common expressions used in the exams. 

Word Statement

Example 

Add / the sum of Sum of 2x and 6=2x+6
Subtract / the difference between Subtract y from 8=8-y
Multiply / the product of Product of 5x and 7=5×7=35x
Divide / the quotient of Quotient of 10y and 2=10y2=5y
Square of / Square root of Square of 3y=3y2=9y2
Cube of / Cube root of Cube root of 8×3=38×3=2x

 

Important Algebraic Notations

The following table outlines key algebraic notations that every student should familiarise themselves with.

Notations

 

Important Algebraic Notations: Distributive Law

This law allows us to simplify and expand algebraic expressions by distributing terms across brackets. The table below outlines key notations related to the distributive law, understanding its application will empower you to efficiently solve equations and manipulate expressions with confidence. 

 

Algebraic Distribution Law

 

Application Questions

A Brief Recap: Negative & Positive Terms 

Before tackling application questions, can you recall how to perform the four operations with positive and negative terms?

Different signs

 

Let’s delve deeper into applying the concepts we’ve learned so far! Test your understanding of algebraic expressions by answering the following questions.

 

Checkpoint! Try to answer the following question.

 

Algebraic Expressions: Question 3

Simplify each of the following.

  1. -5(p-18q)-4(3q-8p)
  2. 124q-8(-q+9p-6q)
  3. 6xy-11x(-2y+3)-2y(-7x-z)

 

Check your Answers! Did you get it right?

Answer 3

 

Factorising Algebraic Expression

Factorisation is the process of expressing an algebraic expression as the product of at least two factors. It is the reverse of expansion.

Algebraic FE

 

Checkpoint! Try to answer the following question.

Factorisation Question 1

Factorise the following:

  1. -8xyz-2xy-20xy2
  2. 125p2q3-60pq2-5pq2
  3. b(4p+4q)-8b(5p-1)

 

Check your Answers! Did you get it right?

FQ1

 

Algebraic Linear Equations and Applications

An equation is a statement where two mathematical expressions have equal values.

 

Algebraic Linear Equations

To solve algebraic linear equations, such as 2x-[3+3(x-8)]=6, perform the following steps.

Steps

Example Workings

  1. Simplify the given equation where possible.
    • Remove brackets.
2x-[3+3(x-8)]=6

2x-3-3x+24=6

  1. Group like terms together on the same side of the equation.
2x-3x=6-24+3
  1. Solve for the unknown
-x=-15

x=15

 

Algebraic Linear Equations with Fractions

To solve algebraic linear equations with fractions, such as 2x-33-3×5=12, perform the following steps.

Steps

Example Workings

  1. Simplify the given equation where possible. 
    • Remove brackets, combining fractions together. 
    • Cross multiplication can be applied if the equation is in the format of fraction = fraction.
2x-33-3×5=12

5(2x-3)-3(3x)15=12

10x-15-9×15=12

x-1515=12

  1. Group like terms together on the same side of the equation.
2(x-15)=15

2x-30=15

  1. Solve for the unknown.
2x=15+30

2x=45

x=2212

 

Cross Multiplication in Algebraic Expressions

When faced with algebraic equations involving fractions, cross-multiplication provides a powerful method to simplify and solve for unknown variables. Let’s delve into this method and explore how it can be applied to solve various types of equations, such as the following.

 

To solve algebraic linear equations with fractions using cross multiplication, perform the following steps. 

Steps

Example Workings

  1. Check if the format of the equation is Fraction = Fraction.

*Must be only a single fraction on each side.

32x-4=45x-3
  1. Left-hand-side numerator multiply to right-hand-side denominator. 
  2. Right-hand-side numerator multiply to left-hand-side denominator.
  3. And equate them.
3(5x-3)=4(2x-4)
  1. Simplify and solve.
15x-9=8x-16

7x=7

x=-1

 

Algebraic Application Questions

Now, let’s explore exam-style questions involving algebraic expressions.

 

Let’s solve the first question together!

 

Question: It is given that the number of a fraction is 3 less than its denominator. If 5 is added to both the numerator and the denominator, the new fraction is equivalent to 34. Find the original fraction.

Steps

Example Workings

  1. Let x be the unknown variable which is required to be found (if the question did not already do so).
Let x be the denominator.
  1. Use the information in the question to form an equation involving the variable x.
x-3+5x+5=34

x+2x+5=34

  1. Solve the equation in x and use it to answer the question.
4(x-2)=3(x+5)

4x+8=3x+15

x=7

Original Fraction =7-37=47

 

Now, it’s your turn!

 

Checkpoint! Try to answer the following question.

Algebraic Application Question 1

Aiden’s father is 11 years older than 3 times his age. Aiden’s mother is 9 years younger than Aiden’s father. Let f, m, and a be the present age of Aiden’s father. Aiden’s mother and Aiden respectively.

  1. Express  f and m in terms of a respectively.
  2. If a=15, find the age of Aiden’s mother in 6 years’ time.

 

Check your Answers! Did you get it right?

Algebraic q

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