Prove algebraically

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GCSE Maths Algebra Proof Maths

Prove Algebraically

Here we will learn about proving algebraically, including proving expressions involving integers that are even, odd or multiples of other numbers, and proving expressions with positive or square numbers.

There are also prove algebraically worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What does prove algebraically mean?

Prove algebraically is a phrase used when you need to prove a statement involving integers, a problem involving algebraic terms, or prove an identity.

You will be used to seeing equations in maths, written with an equals sign $(=),$ such as in the equation $x+2=5.$ An equation is only true for certain values of the variable – here, the statement $x+2=5$ is only true for $x=3.$

An identity is true for any value of the variable. For example, $2x+4x$ is always equal to $6x,$ no matter what values of $x$ are chosen, so you could write this using the identity symbol $(\equiv)$ as $2x+4x \equiv 6x.$

In order to prove that the sum of any two odd numbers is an even number, you cannot just pick a few numerical examples and show it works,

1+3=4, \ 5+7=12.

This is because you need to show that it works for every pair of odd numbers, instead of just the examples you’ve chosen.

To do this you need to know how to express different types of integers algebraically.

You can let any integer be represented by a letter. You can use any letter you want.

For algebraic proof we tend to use the letter $n$ and may also use $m$ if we need a second letter, but you can also use $x, y, w,$ etc.

Even numbers

If you want to express an even number you can call it $2n,$ because all even numbers are multiples of $2.$ A multiple of $3$ would be $3n,$ a multiple of $4$ would be $4n$ etc..

Odd numbers

The odd numbers are always $1$ above or below an even number. So you could call an odd number $2n+1.$

Consecutive integers

Consecutive means “one after the other”.

If you let $n$ be any integer, the next integer will be $n+1.$ The one before $n$ will be $n-1$

$… , n-2, \ n-1, \ n, \ n+1, \ n+2, ...$ are consecutive integers.

$… , 2n-4, \ 2n-2, \ 2n, \ 2n+2, \ 2n+4, ...$ are consecutive even numbers.

$… , 2n-5, \ 2n-3, \ 2n-1, \ 2n+1, \ 2n+3, ...$ are consecutive odd numbers.

Let’s look at an example.

Prove that any two odd numbers add to an even number.

You need to have two different algebraic odd numbers. For this you should use two different letters so that the odd numbers are not related.

Let’s use $2n+1$ and $2m+1.$

Adding these together gives $2n+1+2m+1=2n+2m+2.$

To show that this is even, you just need to show that this expression is a multiple of $2.$

To do that, show you can remove a factor of $2$ from each term.

2n+2m+2\equiv 2\left( n+m+1 \right)

So because you can factor out $2,$ it means the expression must be even.

Therefore, you have proved algebraically that the sum of two odd numbers is always even.

Step-by-step guide: Mathematical proof

What does prove algebraically mean?

How to prove algebraically

In order to prove algebraically:

Think about what algebraic expression will prove the given statement.
Create an expression or manipulate a given expression.
Use a method of simplification, factorising, completing the square or otherwise to find an expression to prove the given statement.

Explain how to prove algebraically

Prove algebraically worksheet

Get your free prove algebraically worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Prove algebraically worksheet

Get your free prove algebraically worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Prove algebraically examples

Example 1: prove an identity

Prove that ${{\left( 5n+2 \right)}^{2}}-{{\left( 4n+5 \right)}^{2}}\equiv {{\left( 3n-4 \right)}^{2}}+4n-37$

Think about what algebraic expression will prove the given statement.

You need to expand the brackets on the left and collect terms. You then need to separate the terms to give the expression equivalent to ${{\left( 3n-4 \right)}^{2}}+4n-37$

{{\left( 3n-4 \right)}^{2}}\equiv 9{{n}^{2}}-24n+16

2Create an expression or manipulate a given expression.

Expand the left hand side.

\begin{aligned} {{\left( 5n+2 \right)}^{2}}-{{\left( 4n+5 \right)}^{2}}& \equiv \left( 25{{n}^{2}}+20n+4 \right)-\left( 16{{n}^{2}}+40n+25 \right) \\\\ & \equiv 9{{n}^{2}}-20n-21 \end{aligned}

3Use a method of simplification, factorising, completing the square or otherwise to find an expression to prove the given statement.

Use the expanded expression for ${{\left( 3n-4 \right)}^{2}}\equiv 9{{n}^{2}}-24n+16$ to find the remaining required terms.

\begin{aligned} 9{{n}^{2}}-20n-21 & \equiv 9{{n}^{2}}-24n+16+4n-37 \\\\ & \equiv {{\left( 3n-4 \right)}^{2}}+4n-37 \end{aligned}

Example 2: prove that an algebraic expression is a multiple of an integer

Prove that ${{\left( 2n+3 \right)}^{2}}-{{\left( n-3 \right)}^{2}}$ is a multiple of $3,$ where $n$ is an integer.

Think about what algebraic expression will prove the given statement.

To prove that the expression is a multiple of $3,$ you need to be able to remove a factor of $3$ from the result.

You need to write the result in the form $3(...).$

Create an expression or manipulate a given expression

Expand the brackets and collect terms.

$\begin{aligned} {{\left( 2n+3 \right)}^{2}}-{{\left( n-3 \right)}^{2}} & \equiv \left( 4{{n}^{2}}+12n+9 \right)-\left( {{n}^{2}}-6n+9 \right) \\\\ & \equiv 3{{n}^{2}}+18n \end{aligned}$

Use a method of simplification, factorising, completing the square or otherwise to find an expression to prove the given statement.

$3{{n}^{2}}+18n\equiv 3\left( {{n}^{2}}+6n \right).$ Therefore, it is a multiple of $3.$

Example 3: prove a worded statement algebraically

Prove that the difference of two consecutive square numbers is odd.

Think about what algebraic expression will prove the given statement.

An odd number can be written in the form $2(...)+1, \ 2(...)-1$ etc., so you need to write your final answer in this form.

Create an expression or manipulate a given expression

Begin by writing the expressions for two consecutive integers $n$ and $n+1.$

The expressions for two consecutive square numbers are just these expressions squared, ${n}^2$ and ${\left(n+1\right)}^2.$

So the difference of two consecutive squares integers can be written as, ${{\left( n+1 \right)}^{2}}-{{n}^{2}}.$

Use a method of simplification, factorising, completing the square or otherwise to find an expression to prove the given statement.

\begin{aligned} {{\left( n+1 \right)}^{2}}-{{n}^{2}} & \equiv {{n}^{2}}+2n+1-{{n}^{2}} \\\\ & \equiv 2n+1. \quad \end{aligned}

This is an odd integer.

Example 4: prove a worded statement algebraically

Prove that the sum of four consecutive odd numbers is a multiple of $8.$

Think about what algebraic expression will prove the given statement.

A multiple of $8$ can be written in the form $8(...).$

Create an expression or manipulate a given expression

The expression for four consecutive odd numbers can be

$2n-3, \ 2n-1, \ 2n+1, \ 2n+3,$ where $n$ is an integer.

The sum of four consecutive odd numbers is

$2n-3+ 2n-1+ 2n+1+ 2n+3.$

Use a method of simplification, factorising, completing the square or otherwise to find an expression to prove the given statement.

2n-3+2n-1+2n+1+2n+3\equiv 8n

Therefore, the sum of four consecutive odd numbers is a multiple of $8.$

Common misconceptions

Incorrect expression for odd and even integers

A common mistake is to write $n$ and $n+1$ for even and odd integers. It is important to remember that even numbers are multiples of $2 \ (2n)$ and odd numbers are an even number plus any odd number $(2n-1).$

Errors when expanding squared brackets

For example, when expanding ${\left( 2n-1 \right)}^{2}$ remember that this means $(2n-1)(2n-1),$ and not just to square $2n$ and square $-1.$

Errors with signs when expanding squared brackets

For example, when expanding $(n-1)^{2},$ remember that ${\left( -1 \right)}^{2}=1$ and not $-1.$

Practice prove algebraically questions

n+1

2n+1

n-1

2n

An odd number is one more or less than a multiple of $2.$

2n, \ 2n+2

n, \ n+2

2n-1, \ 2n+1

n-2, \ n

An even number can be written as $2n.$ The next even number will be $2$ more.

2{{n}^{2}}+1

2{{n}^{2}}+2n

2{{n}^{2}}+2n+1

{{n}^{2}}+n+1

{{n}^{2}}+{{\left( n+1 \right)}^{2}}\equiv {{n}^{2}}+{{n}^{2}}+2n+1

Even

A multiple of $4$

A multiple of $3$

Odd

\begin{aligned} {{n}^{2}}+{{\left( n+1 \right)}^{2}} & \equiv 2{{n}^{2}}+2n+1 \\\\ & \equiv 2\left( {{n}^{2}}+n \right)+1 \end{aligned}

Therefore, the sum of two consecutive square numbers is odd.

Always negative

Always positive

Always odd

Always even

${{\left( n-3 \right)}^{2}}\ge 0, \ {{\left( n-3 \right)}^{2}}+1\ge 1,$ therefore the original expression will always be positive. This is because the square of any (real) number is positive.

Even

Prime

Odd

A multiple of $3$

n+n+1=2n+1

Prove algebraically GCSE questions

1. Prove that the difference between the squares of two consecutive integers is equal to the sum of the two integers.

(3 marks)

Show answer

Expressions for two consecutive integers in the form $n, \ n+1.$

(1)

The expression for the difference of the square given ${{\left( n+1 \right)}^{2}}-{{n}^{2}}.$

(1)

Evidence that the difference is equal to $2n+1.$

(1)

2. (a) Prove the identity

{{\left( 3n+1 \right)}^{2}}-{{\left( 3n-1 \right)}^{2}}\equiv 12n.

(b) Hence, prove that ${{\left( 3n+1 \right)}^{2}}-{{\left( 3n-1 \right)}^{2}}$ is always a multiple of $4$ for positive integer values of $n.$

(4 marks)

Show answer

(a)

$9{{n}^{2}}+6n+1$ or $9{{n}^{2}}-6n+1$

(1)

Attempted subtraction shown.

(1)

Evidence of $12n$ found.

(1)

(b)

$12n=4(3n),$ and statement given.

(1)

3. Prove that the difference between any two even square numbers will be a multiple of $4.$

(3 marks)

Show answer

$2n$ and $2m$ .

(1)

4{{n}^{2}}-4{{m}^{2}}

(1)

4\left( {{n}^{2}}-{{m}^{2}} \right)

(1)

Learning checklist

You have now learned how to:

Understand the difference between an equation and an identity

Argue mathematically to show algebraic expressions are equivalent

Use algebra to support and construct arguments and proofs

The next lessons are

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Introduction

What does prove algebraically mean?

How to prove algebraically

Prove algebraically worksheet

Prove algebraically examples

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Example 1: prove an identity Example 2: prove that an algebraic expression is a multiple of an integer Example 3: prove a worded statement algebraically Example 4: prove a worded statement algebraically

Common misconceptions

Practice prove algebraically questions

Prove algebraically GCSE questions

Learning checklist

Next lessons

Still stuck?