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This page sets out the method for forming & solving 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 Forming & solving equations topic page.
When a question describes lengths, angles or amounts in terms of a letter and gives a total, build an equation.
When there is no letter given, invent one for the smallest or simplest quantity.
Let n be the number you are asked to find.
Say that in a short line before the equation.
“Double a number, then add 5, gives 17” is 2n + 5 = 17.
The word “gives” or “is” becomes the equals sign.
Undo the operations in reverse order.
Subtract 5, then divide by 2. n = 6.
If one person has n and the other has twice as many, the second is 2n.
Their total is n + 2n = 3n.
A rectangle of width n and length n + 3 has perimeter 2n + 2(n + 3).
Set that equal to the given perimeter.
If the question asked for the longer side, do not stop at n.
Substitute back and write the side they asked for.
Write one line that defines the letter before any algebra. "Let the smaller number be n" is enough. Then build the rest from it. "3 more than twice the smaller" is 2n+3, not 3-2n. The phrase "more than" adds onto the thing it follows. Reading it backwards is the same slip as writing "5 less than n" as 5-n.
If two people or two lengths are involved, define one and express the other. Do not invent a second letter unless the question has two independent unknowns, which is simultaneous equations. One letter keeps the equation linear and solvable with the balance method.
Method: 1. Define the letter in one sentence before you write any algebra. 2. “Let the smaller number be n” is enough of a definition. 3. Build every other quantity in the story from that letter.
A perimeter question is an addition of every side, set equal to the given distance. A rectangle has four sides, so two lengths and two widths. If the length is 3x and the width is x+2 and the perimeter is 44, the equation is 2(3x)+2(x+2)=44. Adding only one of each side produces an equation about a different shape.
An angle question uses the angle sum you are told, or 180^° for a triangle and for a straight line. Write each angle as an expression, add them, and set the total equal to 180. "The diagram looks like a right angle" is not a fact unless a square is marked or the question says so. Only marked or stated facts go into the equation.
Method: 1. For a perimeter, add every side and set the total equal to the given distance. 2. Count two lengths and two widths when the shape is a rectangle. 3. With length 3x, width x+2 and perimeter 44, write 2(3x)+2(x+2)=44.
Solve the equation with one operation per line. Then read the question again. "Find the value of x" stops at x=ldots. "Find the largest angle" needs that expression evaluated. "Find the length and the width" needs both. A correct x with the wrong final quantity is a lost accuracy mark, not a complete answer.
Check in the story, not only in the simplified equation. If the angles were x, 2x+10 and 3x-10 and you found x=30, the angles are 30^°, 70^° and 80^°. They add to 180, and the largest really is the one you called largest. A length that comes out negative is rejected. A length that comes out longer than the perimeter it sits inside is a sign error in the equation, and it is worth finding before you box the answer.
Method: 1. Solve the equation with one operation on each line. 2. Read the question again once you know the value of the letter. 3. Stop at the letter only when that is all the question asks for; otherwise work out the lengths or angles it names.
Exam problems hide an equation in a sentence. Start by choosing a letter and saying what it stands for, such as "let n be the number". Then translate the sentence in order. "Zara multiplies a number by 3 and adds 8, and the result is 41" gives 3n+8=41. The word "result" or "is" marks the equals sign. If two operations are linked by a "then", the first one is done first.
Brackets are needed when an operation applies to a whole step. "Add 6 to a number and double it" is 2(n+6)=30, not 2n+6=30. Compare that with "double a number and add 6", which is 2n+6. Check your equation by trying a simple number in the words and in the symbols. If both give the same result for n=5, the equation matches the sentence.
Method: 1. Write "let n be..." so that the letter has a clear meaning. 2. Translate each phrase in order and decide where brackets are needed. 3. Write the equals sign where the sentence says "is", "gives" or "the result is". 4. Test the equation with a simple value to see that it matches the sentence.
Tip: Say what the letter stands for in your answer. It is often worth a mark of its own.
Geometry gives the fact that makes the equation. The angles in a triangle add up to 180^°, angles on a straight line also total 180^°, angles in a quadrilateral add up to 360^°, and the perimeter is the sum of all the sides. For a triangle with angles x, (x+30) and (2x-10) the equation is x+(x+30)+(2x-10)=180, which simplifies to 4x+20=180 and gives x=40.
Do not stop at x. The question usually asks for the size of an angle or a length, so substitute back into each expression. Here the angles are 40^°, 70^° and 70^°. A good check is that the three angles add to 180^°. For two shapes with the same perimeter, write an expression for each perimeter and set them equal to each other.
Method: 1. Write the geometrical fact in words, such as "angles in a triangle add up to 180". 2. Put the expressions into the fact, using brackets, and simplify. 3. Solve for x, then substitute to find the quantities asked for. 4. Check the fact is satisfied, for example that the angles total 180^°.
Many questions describe two or more people or items linked by a rule. Let one quantity be n and write the others in terms of n. If an adult ticket costs £3 more than a child ticket and the child ticket is £c, then an adult ticket is £(c+3). Two adult and three child tickets cost 2(c+3)+3c=5c+6. Set this equal to the total to find c.
Consecutive whole numbers are written n, n+1 and n+2. Three consecutive numbers that add to 48 give n+(n+1)+(n+2)=48, so 3n+3=48 and n=15. The numbers are 15, 16 and 17. Age problems use the same pattern: if Ali is a years old, then his sister who is 4 years older is a+4, and a mother three times as old as Ali is 3a.
Method: 1. Pick one unknown and write every other quantity in terms of it. 2. Add the expressions to match the total in the question. 3. Solve the equation and substitute back to find all the values asked for. 4. Check that the values add up to the given total and satisfy every sentence in the question.
After solving, ask whether the answer makes sense. A number of people, tickets or sandwiches must be a whole number and cannot be negative. A length cannot be negative. If your solution is x=-2 for a length, the equation was probably set up wrongly. Re-read the sentence, check each expression and look at the signs.
Always answer the question that was asked. If it asks for the width, work out the width from x. If it asks for the area, use the length and width you have found. Write a final line with units, such as "the length is 13 cm". Then check by putting your values into the original sentence and making sure every statement is true. This step often finds a slip before the examiner does.
Method: 1. Solve the equation carefully, one line at a time. 2. Decide what the question really asks for, such as a side, an angle or an age. 3. Substitute the solution into each expression to find those values. 4. Check them against the original sentence and write units in the answer.
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