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This page sets out the method for simplifying algebra in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Simplifying algebra topic page.
When you see an expression with + and − between terms, group the same kinds of term together.
When you see × or ÷ (or a fraction) between algebraic terms, deal with numbers first, then letters.
4x and 3x are like terms. 4x and 3 are not.
You can add like terms. You cannot merge a term with a constant.
The coefficient is the number in front of the letter.
In 4x the coefficient is 4. A plain x has coefficient 1.
4x + 3x = 7x. The letter stays.
5x − 2x = 3x. Do not write 3x².
In 4x + 3 + 2x + 1, collect x terms and collect numbers.
That is 6x + 4.
2a + 3b + a = 3a + 3b.
a and b stay in separate terms.
x + x = 2x, not x².
x² would be x times x.
Circle each term with the sign that sits in front of it. In 6a+5b-2a+b the terms are +6a, +5b, -2a and +b. The last b has no written number, so its coefficient is +1. Group the a terms and the b terms on their own. 6a-2a=4a and 5b+b=6b, so the expression is 4a+6b. Stop there. 4a and 6b are unlike, and unlike terms are already simplified.
A power is part of the identity of a term. 2x^2+3x cannot become 5x or 5x^2. The square counts as a different kind of object, in the same way that 2 metres and 3 centimetres are not 5 of the same unit. Writing the higher power first is a tidy habit, but any order of terms is correct. Do not write 1x or x^1. Write x.
Method: 1. Keep the sign in front of a term attached to that term. 2. Split 6a+5b-2a+b into the four signed terms +6a, +5b, -2a and +b. 3. A letter with no written number has coefficient 1, so the last term is +1b.
A number written next to a letter means multiply. So 3p× 4q is 3× 4× p× q=12pq. Letters are written in alphabetical order, so qp is tidied to pq. Repeated multiplication of the same letter builds a power: a× a× a=a^3. Repeated addition is different: a+a+a=3a. Mixing those two up loses the mark, because a power and a coefficient do different jobs.
When a product already contains a power, add the powers of the matching letter. 2x× 3x=6x^2, because x× x=x^2. 4x^2× 3x=12x^3. The numbers multiply as ordinary numbers. A minus sign in front of either factor makes the product negative: -2x× 5y=-10xy. Two minuses make a plus.
Method: 1. Read a coefficient beside a letter as multiplication, not addition. 2. Multiply the numbers and the letters separately, so 3p× 4q becomes 12pq. 3. Finish by writing the letters in alphabetical order.
Division undoes multiplication, one piece at a time. 12m÷ 3=4m, because 12÷ 3=4 and the m has nothing to cancel with. 6ab÷ 2b=3a, because 6÷ 2=3 and one b cancels. You may only cancel a letter that appears in both the top and the bottom. 8a÷ 2b=4a/b, which is 4a/b. The a stays on the top.
A mixed expression is simplified in two passes. First rewrite every product or quotient as a single term. Then collect any like terms that are left. For 3x× 4x-5x^2 you first get 12x^2-5x^2, and only then collect to 7x^2. Doing both jobs at once is how signs get lost.
Method: 1. Cancel one factor at a time when you divide terms. 2. Divide the numbers in 12m÷ 3 to leave 4m, since nothing cancels the m. 3. Cancel a shared letter as well: 6ab÷ 2b leaves 3a.
Real exam questions often hide the like terms inside brackets. When a bracket has a plus sign in front you can simply remove it. When it has a minus sign in front, every term inside changes sign. So 7x-(3x+4) becomes 7x-3x-4, which is 4x-4. A bracket in front of a number, such as 3(2x+1), is expanded by multiplying every term by that number.
The classic slip is changing only the first term inside the bracket. In 9y-(4y-5) the -5 becomes +5, giving 9y-4y+5=5y+5. Write the expanded line first, then collect like terms on the next line. If a question says "Simplify fully", you must finish with no brackets and no two terms of the same kind left separate.
Method: 1. Remove or expand every bracket first, changing signs inside after a minus. 2. Rewrite the whole expression with no brackets. 3. Collect the x terms, then the y terms, then the numbers, keeping each sign with its term. 4. Check by substituting a small number, such as x=2, into the start and the end.
Tip: A minus in front of a bracket is the same as multiplying the bracket by -1.
A term with a^2 is a different kind of term from one with a. You can add 3a^2 and 5a^2 to get 8a^2, but 3a^2+5a cannot be shortened. In a longer expression such as 4n^2+6n+2-n^2+5n-9 you collect three separate groups: the n^2 terms, the n terms and the plain numbers.
Exam papers set this as a three or four mark item because there are three answers to find. Underline or circle each group in a different way so you do not lose a term. Write the answer in a tidy order, highest power first: 3n^2+11n-7. Do not forget that -n^2 is the same as -1n^2, so 4n^2-n^2=3n^2.
Method: 1. Mark the n^2 terms, the n terms and the numbers in three different ways. 2. Add the coefficients of the n^2 terms, remembering a bare n^2 has coefficient 1. 3. Add the coefficients of the n terms, then the numbers. 4. Write the answer with the highest power first.
Tip: The powers on the letters never change when you add or subtract like terms. Only the numbers in front change.
A fraction bar means divide everything on top. In (10v+15)/5 both terms are divided by 5, so the result is 2v+3. The mistake that appears again and again is dividing only the first term and leaving 2v+15. A multiple choice version of this question lists exactly that wrong option, so check both terms before you circle.
You can also go the other way round to check. Multiply your answer back out: 5(2v+3)=10v+15, which matches the top of the fraction. If the number on top will not divide exactly, the expression cannot be simplified this way and you should look again at the question. Fractions of this kind need every term on the top to be a multiple of the bottom number.
Method: 1. Check that every term on the top divides exactly by the number on the bottom. 2. Divide each term separately, keeping its sign. 3. Write the new expression on one line. 4. Multiply it back out to test that you return to the original top.
Two command styles are common. "Show that" gives you the answer, so every line of working must be there and the last line must match exactly. "Is Tom correct?" asks you to find the true simplified form and then compare. State clearly "No" or "Yes" and give the correct expression. A bare "No" with no working earns nothing.
A quick test catches most slips. Choose a simple number such as x=1 and y=2 and put it into the original expression and into your answer. If the two values differ, there is a mistake. For example 2x+3y+4x-y with x=1 and y=2 equals 2+6+4-2=10 and 6x+2y equals 6+4=10, so the match supports the answer.
Method: 1. Expand brackets and collect like terms on separate lines. 2. End the working with the exact form the question asks for. 3. For "Is it correct?" write the true answer and then say Yes or No. 4. Test with a small number, using the same number in the start and the end.
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