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?).
Examples of basic expressions include:
- Linear Expression: 3x+5
- Quadratic Expression: 2×2-4x+7
- 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.
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.
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?
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.
- -5(p-18q)-4(3q-8p)
- 124q-8(-q+9p-6q)
- 6xy-11x(-2y+3)-2y(-7x-z)
Check your Answers! Did you get it right?
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.
Checkpoint! Try to answer the following question.
Factorisation Question 1
Factorise the following:
- -8xyz-2xy-20xy2
- 125p2q3-60pq2-5pq2
- b(4p+4q)-8b(5p-1)
Check your Answers! Did you get it right?
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 |
|
2x-[3+3(x-8)]=6
2x-3-3x+24=6 |
|
2x-3x=6-24+3 |
|
-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 |
|
2x-33-3×5=12
5(2x-3)-3(3x)15=12 10x-15-9×15=12 x-1515=12 |
|
2(x-15)=15
2x-30=15 |
|
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 |
*Must be only a single fraction on each side. |
32x-4=45x-3 |
|
3(5x-3)=4(2x-4) |
|
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 |
|
Let x be the denominator. |
|
x-3+5x+5=34
x+2x+5=34 |
|
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.
- Express f and m in terms of a respectively.
- If a=15, find the age of Aiden’s mother in 6 years’ time.
Check your Answers! Did you get it right?
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