Paper Together: sit a paper in a room with your friends.See how it works
This page sets out the method for substitution 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 Substitution topic page.
When you see 'work out the value of … when …', copy the expression and swap every letter for its number in brackets.
When more than one letter is given, or a value is negative, use a bracket for every replacement.
If x = 4, 3x + 2 = 3 × 4 + 2.
The multiply happens before the add, so the value is 14.
x means 1x.
If x = 5, x + 3 = 8.
If x = 3, x² + 1 = 9 + 1 = 10.
Only the x is squared, unless a bracket says otherwise.
If a = 2 and b = 5, 3a + b = 6 + 5 = 11.
Substitute both before you add.
If x = −2, 3x = −6.
A negative times a positive is negative.
Write the expression with numbers in place of letters, then calculate.
Do not drop the coefficient.
Write the expression again with every letter replaced and every multiplication sign shown. For 4x+3 when x=5, the line is 4× 5+3. Multiply before you add: 20+3=23. The same habit fixes 5+2× 6. The product is 12, then 5+12=17. Adding first to get 42 answers a different expression, (5+2)× 6.
A square applies only to the letter unless a bracket says otherwise. 2x^2 when x=3 is 2× 3^2=2× 9=18. It is not (2× 3)^2=36. If you want the whole product squared, the expression would have been written (2x)^2. Read the position of the power before you press a square key.
Method: 1. Copy the expression with each letter replaced and each multiplication sign written out. 2. When x=5, replace 4x+3 with 4× 5+3. 3. Do the multiplication first: 20+3=23.
Put a negative value in brackets as soon as you substitute. x=-2 into x^2 is (-2)^2=(-2)×(-2)=4. The same value into -x^2 is -(-2)^2=-4, because the minus is outside the square. On a calculator, the brackets are part of the typing. Some calculators read -2^2 as -(2^2), which gives -4, the second of those two results.
A product of two negatives is positive. In 3xy with x=-2 and y=-4, the product xy=(-2)×(-4)=8, then 3× 8=24. A minus already written in the expression stays in front of that product. So 5x^2-3xy becomes 5× 4-3× 8=20-24=-4. Do one piece per line. Combining the signs in your head is where a plus turns into a minus.
Method: 1. Bracket a negative value at the moment you substitute it. 2. Substituting x=-2 into x^2 means (-2)^2=4. 3. The same value in -x^2 keeps the minus outside the square, so the result is -4.
A formula is an equation you are allowed to trust. You do not rearrange it unless you are asked for a different letter. To find v from v=u+at, substitute the three given values. With u=8, a=-3 and t=5, write v=8+(-3)× 5. Multiply first: (-3)× 5=-15. Then v=8-15=-7. The negative acceleration has reduced the speed, which is what the story should lead you to expect.
A longer formula such as s=ut+1/2at^2 uses the same order. Square t before you multiply by 1/2a. With u=6, a=4 and t=3, s=6× 3+1/2× 4× 3^2=18+2× 9=18+18=36. Keep the fraction until you have multiplied it by the squared term. Halving 4 first, to get 2, is fine, because 1/2× 4=2. Halving after you have added the ut term is not.
Method: 1. Treat a formula as an equation you may use as written. 2. Rearrange only when a different letter is the one you have been asked for. 3. To find v from v=u+at, substitute the three numbers you were given.
Substitution means replacing each letter with its number and then working out the value. The order of operations still applies after you have replaced the letters, so powers come before multiplying and multiplying comes before adding. In the formula T=5P-W with P=4 and W=2 you work out 5×4=20 first, then 20-2=18. A joined pair of letters such as ab always means a× b.
A neat habit is to rewrite the expression with brackets around each number before you calculate. For 4n^2-3m with n=-3 and m=-2 write 4×(-3)^2-3×(-2). That one line shows the examiner your method and prevents sign slips. The square belongs to the number n only, not to the 4, so (-3)^2=9 and the first term is 36.
Method: 1. Rewrite the expression with every letter replaced by its number in brackets. 2. Work out powers first, then multiplication and division. 3. Do additions and subtractions last, left to right. 4. Write the answer with the unit if the formula has one.
Tip: When a value is negative always keep it in brackets while you substitute. It is the best protection against sign errors.
Two negatives next to each other cause most of the lost marks. Subtracting a negative is the same as adding: a-b with a=3 and b=-5 is 3-(-5)=8. A negative number squared is positive, (-4)^2=16, but -4^2 on its own is -16 because only the 4 is squared. Say the sign out loud as you write each line.
On a calculator paper use the bracket keys for negative numbers and fractions. Type (-4)^2 with brackets and put brackets around a whole top line before you press divide. Check with an estimate: if you expect an answer of about 50 and the calculator shows 5000 you have probably missed a bracket. On a non-calculator paper the numbers are friendly, so work each stage on paper in a column.
Method: 1. Replace each letter by its number in brackets, including the minus sign. 2. Work out each term separately and write its value. 3. Combine the terms, using the rules for adding and subtracting negatives. 4. On a calculator use brackets for negatives and for fractions, then estimate to check.
Sometimes the answer is not on the left of the formula. You are told the value of the subject and asked for one of the other letters. For R=4S-T with R=-12 and T=8 the substitution gives an equation: -12=4S-8. Add 8 to get -4=4S, then divide by 4 to get S=-1. These questions are worth three marks because of the extra step.
Work in a clear order. Substitute first, then solve the equation you have made. If the letter you want is squared, take the square root at the end, and remember that a physical quantity such as speed is positive. Check your answer by putting it back into the formula with the other given value: 4×(-1)-8=-12, which matches the number in the question.
Method: 1. Substitute the known values and leave the unknown letter. 2. Solve the equation by doing the same to both sides. 3. If the unknown is squared, take a square root at the end. 4. Check by substituting your answer into the formula.
A higher question gives two expressions and says they have the same value for a particular number. For 3n and n^2-k with n=12 you work out 3×12=36 and 12^2-k=144-k. Setting them equal gives 144-k=36 so k=108. This links substitution to solving an equation, and it carries three marks.
A "Show that" question gives you the final value, so you must show every stage. For a ball with s=ut+1/2at^2, u=12, a=-10 and t=2.4 you need 12×2.4=28.8 and 1/2×(-10)×2.4^2=-28.8 separately before adding to get 0. Writing only the answer 0 earns nothing. Include a sentence that states what the result means.
Method: 1. Substitute into each expression separately and simplify each. 2. For same-value questions, write an equation by setting the two results equal. 3. Solve the equation for the unknown letter. 4. For "Show that", write each stage of the calculation and end with the given value.
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