Algebra is the use of letters and symbols in Mathematics. They represent numbers and quantities in situations, such as formulae and algebraic equations. Variables are used - like x, y and z - as well as arithmetic operations like multiplying and dividing.
This guide will focus largely on the basics of how to do algebra without a calculator and what you need to know for KS3. Topics covered include algebra as used in formulae, equations and expressions. We will explain how to solve maths equations by 'cancelling' and the use of operations and variables in algebra. A few simple algebra problems are included. If you need further help, you can find a KS3 Maths tutor.
Algebraic notation presents information in a concise way, making it as simplified as possible.
Let's say we have two variables: a and b. When we multiply these variables together, they are written together: ab represents a x b. It's important to place these variables in alphabetical order, so ba would be incorrect.
When we use algebra in a sentence, this is called an expression e.g. 2a - 3. Every part of an expression is a term, so 2a - 3 has two terms: 2a and 3. A term can be any of the following:
- Coefficient combined with another variable/s - A number/symbol multiplied with a variable e.g. 3 is the coefficient of 3b, which means 3 x b or b x b x b)
- Constant - As the name implies, this is a number/quantity that is fixed and doesn't change. For example, the size of a shoe is constant, while the speed of a car varies.
- Variable - This is an unknown value and is represented by a letter, such as x.
We can use algebraic notation to solve worded problems. It can also be added to diagram annotations to resolve geometrical problems. It's a vital skill in Maths and will come in handy when you learn about later topics, including trigonometry and plotting graphs.
Operators are commonly used in algebra (see "Algebraic operations" below). When we use division in algebra, it is written in the fractional form e.g. 5/10. The dividend (the number being divided) is the numerator (the part of the fraction above the line) and the divisor is the denominator. For example, the division of 8 ÷ 2 will be written as 8/2.
When we're using algebraic notation for multiplying, we always have to put the coefficient before the letter. For example, if we multiply the variable a 4 times, it will be written 4a (4 x a or a x a x a x a) rather than a4.
Algebraic statements (inc. expression and formula)
An algebraic statement can be an expression, an equation, a formula, or an identity. Each of these statements has been explained below in more detail:
- Expression - a set of terms with letters and numbers that are combined using operations (+, -, x, ÷) e.g. 2 x 7 + 4xy - c
- Equation - a mathematical statement where 2 expressions are equal to each other e.g. 3x - 2 = 8
- Formula - this is an equation or rule written using symbols and terms. It uses algebraic expressions on both sides of the equals sign and links 2 or more variables. This is a common algebraic formula: (a + b)2 = a2 + 2ab + b2
- Identity - The left half of the equation is equal to the right side for all the values and variables e.g. 2x + 3x = 5x
To correctly calculate an expression, every variable needs to be defined as representing a number of items. We also want to always to use the correct operation inside the expression:
- When an addition is used in an expression, the result will be more e.g. x + 5 means increase x by 5 or 5 more than x.
- When subtraction is used, the result will be less e.g. x - 3 means decrease x by 3 or 3 fewer than x.
- When an amount is multiplied, the result will be this value however many times larger e.g. 7x means 7 times as many as x.
- When an amount is divided, the result will be this value however many times smaller. x/4 means a quarter of x.
- p - 10 means ten fewer than a number (p).
- p / 10 means a tenth of a number (p).
- 8 - p means 8 decreased by a number (p).
- p + 20 means 20 more than a number (p).
- 8p means 8 times as much as a number (p).
What is the correct way to write a x 3 x b
Felix has n cars. Frank has 4 more cars than Felix. How do you write this as an expression?
There are rules that govern the use of addition and multiplication in algebra:
- Commutative rule of addition - When 2 terms are added together, the order they are added doesn't matter. This is expressed as (a + b) = (b + a). For example, (y3 + 3y) = (3y + y3).
- Associative rule of addition - When 3 or more terms are added together, the order they are added doesn't matter. This is expressed as a + (b + c) = (a + b) + c. For example, y3 + (2y3 + 3) = (y3 + 2y3) + 3.
- Commutative rule of multiplication - When 2 terms are multiplied, the order they are multiplied in doesn't matter. This is written as (a × b) = (b × a). For example, both the following equations return the same answer: (y4 - 2y) × 3y = (3y5 - 6y2) and 3y × (y4 - 2y) = (3y5 - 6y2).
- Associative rule of multiplication - When 3 or more terms are multiplied, the order they are multiplied in doesn't matter. The equation is written as a × (b × c) = (a × b) × c. For example, y4 × (3y5 × y) = (y4 × 3y5) × y.
- Distributive rule of multiplication - When we multiply a number by a group of numbers that are added together, this is the same as doing each multiplication separately. This is written as a × (b + c) = (a × b) + (a × c). For example, y2 × (2y + 1) = (y2 × 2y) + (y2× 1).
Algebraic Operations (inc. multiply, subtract and divide)
Arithmetic operations (such as + and -) are used in algebra to simplify expressions, solve equations and rearrange formulae (see "Solve equations by 'Cancelling' below).
When solving equations and expressions, it's important to work through the order of operations in order from left to right. Doing the operations in the wrong order can potentially affect the answer. The acronym PEMDAS is a handy way to remember this order:
- Multiplication or Division
- Addition or Subtraction
Let's apply PEMDAS to the following example: 2 x 6 ÷ (8 - 2) - 23 + 3 x 4:
- Parentheses: 2 x 6 ÷ (6) - 23 + 3 x 4
- Exponents: 2 x 6 ÷ 6 - 8 + 3 x 4
- Multiplication/division: 12 ÷ 6 - 8 + 3 x 4
- Multiplication/division: 2 - 8 + 12
- Addition/subtraction: -6 + 20
- Addition/subtraction: 6
These are the 4 algebraic operations that you will most commonly use in algebra: addition, subtraction, multiplication and division.
We often see letters and symbols when working with algebra. These are called variables, which is how we show numbers with unknown values. We can use any letters, with the following especially common: x, y, a, b, c. We will also see Greek letters, like theta (θ). When we do an equation, a number will go in place of the variable: your job is to discover what mystery number the variable is hiding.
- Let's say we have the equation 5x - 10 = 85. In this equation, x is the variable. To find the value of x, we need to cancel the "-10" by adding 10 to 85 (95). We now reverse the multiplication of x by dividing 95 by 5, which equals 19. Therefore, x = 19.
If you're struggling with the use of letters, it may help to replace them with question marks. This amounts to the same thing and is worth doing if it makes it easier to process. For example 5 + x = 4 can be written 5 + ? = 4 (the final answer is -1).
You may see a variable more than once in an equation. When this happens, it's helpful to think of them as numbers and try to combine them to simplify the equation. You can use operations to do this, such as adding or subtracting them. You wouldn't think twice about simplifying 2 + 3 x y = 8 to 5 x y = 8. The same can be applied to y + y x z = 7, simplifying it to 2y x z = 7.
Remember that you can only add the same variables together. In the equation 3y + 2z = 9, we can't combine 3y and 2z because they are different letters and therefore different variables. This is also true when the same variable has a different exponent e.g. you can't combine 2z and 3z2 in the equation 2z + 3z2 = 8.
How to solve algebra equations by "Canceling"
There are various ways we can solve algebra equations by cancelling. First things first - cancel means to remove or simplify a value/expression by subtracting or dividing with a term. The following example has 8b on both sides of the equals sign: 2a + 8b + 9 = 4a + 8b + 6. We can cancel this term from both sides, by subtracting 8b by 8b. This simplifies the expression: 2a + 9 = 4a + 6.
Listed below are ways we can cancel to solve algebra equations:
- Isolate the variable - To solve an algebra equation normally means we have to work out the value of a variable. There are normally numbers and/or variables on both sides of the equals sign. We need to simplify the equation so the variable is on one side. Take the following equation: y + 4 = 12 x 2. First, we remove "+ 4" from the left side of the expression (y = 12 x 2) and deduct it from the other side to keep both sides equal (y = 12 x 2 - 4). We now multiply 12 x 4 (48) and then subtract 4 (44). This cancels the equation down to y = 44.
- Cancel addition with subtraction (and subtraction with addition) - To get a variable isolated in an equation, we normally need to remove numbers from this side of the equals sign (see above). This means using the opposite operation when moving the term from one side to the other. In the example b - 8 = 2, we need to move "-8" to the other side. First, we switch the operation from subtract to add (+8), which cancels out the 8. This leaves us with b = 2 + 8 = 10. Here is an example where addition is cancelled by subtraction: y + 10 = 5 → y = 5 - 10 = -5.
- Cancel division with multiplication (and multiplication with division) - This is the same idea as above: if you have multiplied or divided on one side of the equals sign, you can cancel it by doing the opposite on the other side. Make sure to multiply/divide everything on the other side of the equals sign. This is a multiplication to division example: 6x = 15 + 3 → x = (15 + 3) ÷ 6 = 18 ÷ 6 = 3. This is a division to multiplication example: x/4 = 7 + 10 → x = (7 + 10) x 4 = 17 x 4 = 68.
- Cancel exponents by taking the root (and take the exponent if you have a root) - A reminder: the exponent is how many times a number is multiplied by itself e.g. in 34 the "4" is the exponent and tells us to multiply 3 by itself 4 times - 3 x 3 x 3 x 3). The opposite of an exponent is the root with the same number. For example, the opposite of the 3 exponent is the cube root (3√) and the opposite of the 2 exponent is a square root (√). If you have an exponent, you need to take the root of both sides and vice versa if you're dealing with a root. when dealing with an exponent. This is an exponent-to-root example: y2 = 21 → y = √21. This is a root-to-exponent example: √x = 7 → x = 72.
A few simple algebra equations are listed to tackle and test your knowledge. Try to solve the equations below.
Solve a + 4 = 2a
Frequently asked questions
The different types of algebraic expression include:
- Monomial - this only has one term e.g 5g
- Binomial - this has two terms e.g. 3a + 5c
- Trinomial - this has three terms and more than 1 variable e.g. 2x + 4m + 7c
- Quadratic - this can be rearranged in standard form, where x is an unknown value and the a, b and c are known numbers. The equation is written ax2 + bx − c. For example, 5x2 + 3x + 4
- Cubic - a cubic equation contains a cubic variable e.g. 8x3 − 3x2 + 5x − 2
The basic four rules in algebra are:
- Associative rule of addition
- Commutative rule of addition
- Associative rule of multiplication
- Commutative rule of multiplication
- Distributive rule of multiplication
When like terms are added or subtracted, the coefficients are added or subtracted and written before the like terms e.g. 2y + 3y = (2+3)y = 5y.
- algebraic notation - Mathematically symbols that are used to represent numbers, amounts or elements.
- coefficient - The number or symbol that is multiplied by a variable in an algebraic term. For example, the coefficient of 3b is 3.
- commutative - The order doesn't matter when using an operation. Multiplication and addition operations are commutative e.g. 4 x 5 = 20 and 5 x 4 = 20.
- constant - A number or quantity that never changes. For example, the number of days in a week is a constant.
- denominator - The number written on the bottom of a fraction - it's the number of equal parts e.g. the denominator in 2/4 is 4.
- equation - A mathematical statement that shows two expressions are equal. Both expressions are linked with the equals sign e.g. a + 2 = 10.
- expression - A mathematical sentence that is written numerically or symbolically and has one or more terms e.g. x-4.
- formula - A fact, rule, or principle expressed with words and mathematical symbols.
- identity - An equation that is true, regardless of the values that are chosen. Identities can be written with the sign ≡ to show the expression is an identity e.g. 4y + 7y ≡ 11y.
- numerator - The number written at the top of a fraction - it's the number of parts used e.g. the numerator in 2/4 is 2.
- term - An element inside an algebraic sentence e.g. 4 + 3y has two terms (4 and 3y). Terms are separated by + or - operations.
- variable - An unknown value that is represented by a letter, such as y.
This post was updated on 15 Aug, 2023.