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This page sets out the method for solving one-step equations 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 Solving one-step equations topic page.
When you see a letter with one operation attached, do the inverse operation to both sides.
When the numbers are small, ask the question in words to check your answer.
An equation says both sides are worth the same.
Whatever you do to one side, do to the other.
In x + 7 = 12, subtract 7 from both sides. That gives x = 5.
In x − 4 = 3, add 4 to both sides. That gives x = 7.
In 4x = 20, divide both sides by 4.
x = 5. Do not subtract 4.
In x ÷ 3 = 6, multiply both sides by 3.
x = 18.
Put your value back into the original equation.
The two sides should match.
These equations have one operation on x.
A later topic deals with two steps, such as 2x + 1 = 9.
In x+9=22 the letter has had 9 added to it. Subtract 9 from both sides: x=22-9=13. Check: 13+9=22. In x-6=11 the letter has had 6 taken away. Add 6 to both sides: x=17. Doing the same operation again, instead of the inverse, is the error that turns x+9=22 into x=31.
A negative solution is allowed. x+11=4 means the letter is smaller than 4. Subtract 11 from both sides: x=4-11=-7. Check: -7+11=4. There is nothing to "fix" about a negative answer. If the check fails, the inverse was the wrong one, or the subtraction was done in the wrong direction.
Method: 1. In x+9=22, notice that 9 has been added to the unknown. 2. Subtract 9 from both sides to reach x=13. 3. In x-6=11, notice that 6 has already been taken away from the unknown.
A number written next to a letter means multiply. 7y=56 means 7× y=56, so divide both sides by 7: y=8. It does not mean 7+y. Reading it as addition gives 49, and 7× 49 is nowhere near 56. The check is immediate: 7× 8=56.
A fraction bar with the letter on top is division. n/4=6 means n÷ 4=6, so multiply both sides by 4: n=24. If the division does not come out as a whole number, give the decimal or the fraction. x/4=2.5 gives x=10, but x÷ 4=10 is a different equation with solution 40. Read which number is the divisor before you multiply.
Method: 1. Read 7y as 7 times y. 2. Divide both sides of 7y=56 by 7 to get y=8. 3. Do not rewrite that equation as 7+y=56.
In 12-x=5 the letter is not the starting amount. Something is being subtracted from 12, and 5 is left. The amount taken away is 12-5=7, so x=7. A useful rearrangement is to add x to both sides, giving 12=5+x, then subtract 5. Either route lands on 7. The wrong sign, x=-7, fails the check because 12-(-7)=19, not 5.
Word problems at this grade are still one inverse, once the equation is written. "A temperature rose by 7 degrees and reached -2" is t+7=-2. "A 15 litre tank has 6.5 litres left after some is poured out" is 15-k=6.5. Write the equation first, solve it, then answer in the unit the story asked for. A bare number with no unit is not the end of a worded question.
Method: 1. In 12-x=5 the unknown is the amount being subtracted, not the starting amount. 2. You begin with 12 and finish with 5. 3. The amount taken away is 12-5, so x=7.
Every one-step equation in an exam is one of four shapes: x+a=b, x-a=b, ax=b or x/a=b. In each case you undo the operation that is attached to the letter. Add to undo a subtract, subtract to undo an add, divide to undo a multiply and multiply to undo a divide. Whatever you do to one side you must do to the other.
Look at what has been done to x before you choose. In x-9=4 the 9 has been taken away, so you add 9 to both sides to get x=13. In 6c=54 the 6 multiplies c, so divide both sides by 6 to get c=9. In d/5=8 the letter is divided by 5, so multiply both sides by 5 to get d=40. Write the new equation on a fresh line each time.
Method: 1. Circle the operation that is joined to the letter. 2. Write down its opposite and apply it to both sides. 3. Write the answer as x= a number on its own line. 4. Put the answer back into the first equation to check.
Tip: If the question says "Solve", write the full line, such as x = 13, and not just 13.
Solutions are not always positive whole numbers. If m+9=3 then m=3-9=-6. A negative answer is perfectly fine and exam questions often use it on purpose. If 4x=10 then x=10÷4=2.5, or 5/2. If the question does not say which form to use, a decimal is usually safe when it ends, and a fraction when it does not.
Do not round an exact answer. For 5y=12 the exact answer is 2.4. If a division does not come out exactly, such as 3k=10, leave the answer as 10/3 unless the question asks for a decimal. Always replace the letter in the original equation as a check: 4×2.5=10 shows the answer is right. Negative checks work the same way, so -6+9=3 confirms the first example.
Method: 1. Undo the operation, writing the result of the arithmetic carefully. 2. Keep the negative sign when a smaller number is taken from a larger one in the wrong order. 3. Leave non-terminating answers as fractions. 4. Substitute your answer back into the original equation.
Equations such as 40-w=27 look different because the letter has a minus sign in front of it. Think of it as "40 take away something is 27". The missing number is the difference, 40-27=13. Alternatively add w to both sides to get 40=27+w and then subtract 27 from both sides. Both routes lead to w=13.
The common slip is writing w=27-40=-13. A quick check kills this: 40-(-13)=53, not 27. Always put your answer back into the equation. This type appears often as a short 1 or 2 mark question on both calculator and non-calculator papers. If the question says "You must show your working", write the inverse step as a line.
Method: 1. Write the equation and read it as a sentence: "40 minus what is 27". 2. Find the missing number with a subtraction, big number minus the result. 3. Substitute your value to make sure the equation is true. 4. Write the answer as w = the number.
Foundation papers often ask you to write an equation and then solve it. "The temperature fell by 6 degrees and was then -1 degrees" becomes t-6=-1. "6 books cost £39 altogether" becomes 6b=39. Choose a letter, write the equation using the information, then solve it. The equation is usually worth one mark and the solution another.
Angles give a second source of one-step equations. Angles on a straight line add up to 180^°, so a line with angles x and 118^° gives x+118=180 and x=62. Always state the unit in the answer and say which angle you found. Check your solution by adding the angles on the line to see that the total is exactly 180.
Method: 1. Pick a letter for the unknown and say what it stands for. 2. Turn the sentence or diagram fact into an equation with an equals sign. 3. Solve the equation by doing the opposite operation on both sides. 4. Answer in the words of the question and include the unit.
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