Chapter 2 of 8

All 8 chapters
  1. 1Harder linear equations, readunderstood
  2. 2Changing the subject, readunderstood
  3. 3Inequalities, readunderstood
  4. 4Equations of straight lines, readunderstood
  5. 5Two equations: adding or taking away, readunderstood
  6. 6Two equations: substitution, readunderstood
  7. 7Two equations on a graph, readunderstood
  8. 8Word problems and regions, readunderstood

M4.8.2 Chapter 2

Changing the subject

The formula v = u + at gives speed. How do I get t on its own?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Rearrange v = u + at to make t the subject
  • Rearrange a formula where the subject appears twice

Tariq rides a bike. The formula v = u + at gives his speed after a time t. Here u is his speed at the start, and a is how much faster he goes each second. The letter v is his speed at the end. Speeds are in metres per second, and time is in seconds.

In v = u + at, the letter v stands alone on the left side. We call it the subject of the formula. What if you know v, u and a, and want t? Then you must rearrange the formula to make t the subject.

Draw it first

Undo the machine

  1. A machine. The row marked forward goes t, × a, at, + u, v. Step 2 adds the row marked undo, which goes v, − u, at, ÷ a, t. Step 3 writes t = (v − u)/a under it.forwardt× aat+ uv

    1 Start with t. Multiply by a to get at. Then add u to get v.

  2. A machine. The row marked forward goes t, × a, at, + u, v. Step 2 adds the row marked undo, which goes v, − u, at, ÷ a, t. Step 3 writes t = (v − u)/a under it.forwardt× aat+ uvundov− uat÷ at

    2 To go back, start at v. Undo the adding by taking u away. Then undo the multiplying by dividing by a.

  3. A machine. The row marked forward goes t, × a, at, + u, v. Step 2 adds the row marked undo, which goes v, − u, at, ÷ a, t. Step 3 writes t = (v − u)/a under it.forwardt× aat+ uvundov− uat÷ att = (v − u)/a

    3 The way back is t = (v − u)/a. The whole of v − u is divided by a.

The undo row goes in reverse order, and each of its steps is the opposite of one step in the forward row. Adding and taking away undo each other. Multiplying and dividing undo each other.

A balance shows why. Take u = 2 and a = 3, so v = 2 + 3t. Say t = 4. Then v = 14. The left pan holds 14 weights. The right pan holds 3 bags of t and 2 weights.

A level balance. The left pan holds 14 weights and is labelled 14. The right pan holds 3 bags marked t and 2 weights, and is labelled 3t + 2.14ttt3t + 2
v = 14 balances u + at = 2 + 3t. Each small square is one unit.

Take u = 2 weights off both pans. That is the same as taking u from both sides. The left pan keeps 12 weights. The right pan keeps the 3 bags.

A level balance. The left pan holds 12 weights and is labelled 12. The right pan holds 3 bags marked t and is labelled 3t.12ttt3t
v − u = 12 balances at = 3t. Each small square is one unit.

Now share the 12 weights between the 3 bags. Each bag is 4. That is the same as dividing both sides by a. You get t = 12 ÷ 3 = 4.

Worked example

Make t the subject of v = u + at.

  1. Start with v = u + at. The letter t is only in the term at.
  2. Take u from both sides: v − u = at.
  3. Divide both sides by a: (v − u)/a = t.
  4. Write t on the left: t = (v − u)/a. This needs a not to be zero, because you cannot divide by zero.
  5. Check with u = 2, a = 3 and t = 4. Then v = 14, and (14 − 2)/3 = 12/3 = 4.

Answer: t = (v − u)/a. With v = 14, u = 2 and a = 3, t = 4.

The same steps work for any formula. The temperature formula F = 1.8C + 32 turns degrees Celsius into degrees Fahrenheit. To go the other way, make C the subject.

Worked example

Make C the subject of F = 1.8C + 32.

  1. The formula multiplies C by 1.8, then adds 32. Undo the adding first.
  2. Take 32 from both sides: F − 32 = 1.8C.
  3. Divide both sides by 1.8: C = (F − 32)/1.8.
  4. Test it. At normal air pressure, water boils at 212 °F, so F − 32 = 180.
  5. Then C = 180 ÷ 1.8 = 100. Water boils at 100 °C at normal air pressure.

Answer: C = (F − 32)/1.8. For F = 212, C = 100.

Sometimes the subject appears twice. In P = ax + bx the letter x is in both terms. You cannot undo one without the other. Instead, factorise: take x out as a common factor.

A rectangle x high and a + b wide. The left cell is ax and the right cell is bx. The width a + b is written over a bracket. The side x is written on the left.abxa + baxbx
ax + bx is a rectangle x high and a + b wide, so ax + bx = x(a + b).

Now x is in one place only. P = x(a + b). Divide both sides by (a + b) and you get x = P/(a + b). This needs a + b not to be zero.

Worked example

Make x the subject of ax + b = cx + d.

  1. Collect the x terms on the left and the other terms on the right. Take cx and b from both sides: ax − cx = d − b.
  2. Factorise the left side. Both terms have x: ax − cx = x(a − c).
  3. Now x is in one place: x(a − c) = d − b.
  4. Divide both sides by (a − c): x = (d − b)/(a − c). This needs a − c not to be zero.
  5. Try a = 5, b = 2, c = 3 and d = 8. Then d − b = 8 − 2 = 6.
  6. And a − c = 5 − 3 = 2.
  7. So x = 6/2 = 3. Put it back: 5 × 3 + 2 = 17, and 3 × 3 + 8 = 17.

Answer: x = (d − b)/(a − c). With a = 5, b = 2, c = 3 and d = 8, x = 3.

See it three ways

With things

Joseph starts a run at 5 metres per second and speeds up by 2 metres per second every second. His speed is now 17 metres per second. How many seconds has he been speeding up?

As a picture

A machine. The row marked forward goes t, × 2, 2t, + 5, 17 m/s. The row marked undo goes 17 m/s, − 5, 2t, ÷ 2, t.forwardt× 22t+ 517 m/sundo17 m/s− 52t÷ 2t

In words

Take 5 away from 17 to get 12. The speed gains 2 every second, so divide 12 by 2. That gives 6 seconds.

In symbols

17 = 5 + 2t, so 2t = 12 and t = 6. Or use the formula: t = (v − u)/a = (17 − 5)/2 = 6.

Try it

Undo a number trick

You need: a friend, paper and a pencil.

  1. Ask your friend to think of a number and keep it secret.
  2. Ask them to multiply it by 3, then add 2, and tell you the answer.
  3. Take away 2, then divide by 3. Tell them their number.

What you will see: You get their number every time. You undid the steps in the opposite order, and each step was undone by its opposite.

If you cannot try it, this is what happens.
A machine. The row marked forward goes 4, × 3, 12, + 2, 14. The row marked undo goes 14, − 2, 12, ÷ 3, 4.forward4× 312+ 214undo14− 212÷ 34

A wrong idea many people have

To get t alone in v = u + at, divide v by a and add u. So t = v/a + u.

Why it feels right: The letter a sits next to t, so it feels as if you can deal with it first.

The test that breaks it: Put u = 2, a = 3 and t = 4, so v = 14. Then v/a + u = 14/3 + 2, which is not 4. But (v − u)/a = 12/3 = 4.

What is true: Undo in reverse order. The u was added last, so take it away first. Then divide the whole of v − u by a.

A wrong idea many people have

You can always divide both sides by a letter.

Why it feels right: Dividing by a number such as 3 always works, so a letter feels the same.

The test that breaks it: Put a = 0 in v = u + at. Then v = u + 0 × t, so v = u whatever t is. Knowing v and u tells you nothing about t.

What is true: You can divide by a letter only when it is not zero. Say so when you need it, as in t = (v − u)/a for a not zero.

Key words

subject
the letter that stands alone on one side of a formula, such as v in v = u + at.
rearrange
to change a formula so that a different letter is the subject.
factorise
to write a sum as a product by taking out a common factor, as ax + bx = x(a + b).

Back to your guess

The answer is B: Take u from both sides, then divide everything by a: t = (v − u)/a.

The formula multiplies t by a, then adds u. To undo it, take away u first, then divide the whole of v − u by a. So t = (v − u)/a.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Make t the subject of v = u + at.

    Tap your answer.

    1. Not this one. You took a away, but a multiplies t. To undo a multiplication, divide. Go to M4.8.2. Try another.

    2. Yes, that is right. Take u from both sides: v − u = at. Divide both sides by a: t = (v − u)/a. With u = 2, a = 3 and t = 4, v is 14 and (14 − 2)/3 is 4.

    3. Not this one. You divided only the v. When you divide one side by a, every term on that side is divided. Take u away first. Go to M4.8.2. Try another.

    4. Not this one. The u is added to at on the right, so you take it away, not add it. Adding u to both sides would not remove it. Go to M4.8.2. Try another.

    See the answer

    t = (v − u)/a

    Take u from both sides: v − u = at. Divide both sides by a: t = (v − u)/a. With u = 2, a = 3 and t = 4, v is 14 and (14 − 2)/3 is 4.

    • If you chose A: You took a away, but a multiplies t. To undo a multiplication, divide. Go to M4.8.2.
    • If you chose C: You divided only the v. When you divide one side by a, every term on that side is divided. Take u away first. Go to M4.8.2.
    • If you chose D: The u is added to at on the right, so you take it away, not add it. Adding u to both sides would not remove it. Go to M4.8.2.

  2. The picture shows w = 3y − 5. Which formula makes y the subject?

    A machine. One row goes y, × 3, 3y, − 5, w.y× 33y− 5w
    Tap your answer.

    1. Not this one. The step in the machine is − 5, so the undo step is + 5. Undoing a minus means adding. Go to M4.8.2. Try another.

    2. Not this one. Undo adds 5 first and divides after. Divide the whole of w + 5 by 3, not just the w. Go to M4.8.2. Try another.

    3. Yes, that is right. Undo the steps in reverse order. The last step is − 5, so add 5 first: w + 5 = 3y. Then undo × 3 by dividing by 3: y = (w + 5)/3.

    See the answer

    y = (w + 5)/3

    Undo the steps in reverse order. The last step is − 5, so add 5 first: w + 5 = 3y. Then undo × 3 by dividing by 3: y = (w + 5)/3.

    • If you chose A: The step in the machine is − 5, so the undo step is + 5. Undoing a minus means adding. Go to M4.8.2.
    • If you chose B: Undo adds 5 first and divides after. Divide the whole of w + 5 by 3, not just the w. Go to M4.8.2.

  3. Make r the subject of C = 2πr, the circumference of a circle.

    Tap your answer.

    1. Yes, that is right. The letter r is multiplied by 2π. Divide both sides by 2π: r = C/(2π).

    2. Not this one. You multiplied by 2π, but r is already multiplied by 2π. To undo it, divide by 2π. Go to M4.8.2. Try another.

    3. Not this one. You took 2π away. The term 2πr is a product, so you undo it by dividing, not by taking away. Go to M4.8.2. Try another.

    See the answer

    r = C/(2π)

    The letter r is multiplied by 2π. Divide both sides by 2π: r = C/(2π).

    • If you chose B: You multiplied by 2π, but r is already multiplied by 2π. To undo it, divide by 2π. Go to M4.8.2.
    • If you chose C: You took 2π away. The term 2πr is a product, so you undo it by dividing, not by taking away. Go to M4.8.2.

  4. The perimeter of a rectangle is P = 2l + 2w. Make w the subject.

    A rectangle with the length l written above it and the width w written on its left side.lw
    Tap your answer.

    1. Not this one. P − 2l is 2w, not w. You must divide by 2 as well. Go to M4.8.2. Try another.

    2. Not this one. The 2 belongs to the l: the term is 2l. Take the whole of 2l away. Go to M4.8.2. Try another.

    3. Not this one. You divided P by 2 but not the 2l. Divide every term by 2, so 2l becomes l. Go to M4.8.2. Try another.

    4. Yes, that is right. Take 2l from both sides: P − 2l = 2w. Divide both sides by 2: w = (P − 2l)/2. This is the same as P/2 − l.

    See the answer

    w = (P − 2l)/2

    Take 2l from both sides: P − 2l = 2w. Divide both sides by 2: w = (P − 2l)/2. This is the same as P/2 − l.

    • If you chose A: P − 2l is 2w, not w. You must divide by 2 as well. Go to M4.8.2.
    • If you chose B: The 2 belongs to the l: the term is 2l. Take the whole of 2l away. Go to M4.8.2.
    • If you chose C: You divided P by 2 but not the 2l. Divide every term by 2, so 2l becomes l. Go to M4.8.2.

  5. Make x the subject of P = ax + bx.

    Tap your answer.

    1. Not this one. Dividing by a + b is not the same as dividing by a and by b. With P = 10, a = 2 and b = 3, x is 2, but P/a + P/b is more than 8. Go to M4.8.2. Try another.

    2. Yes, that is right. Factorise: P = x(a + b). Divide both sides by (a + b): x = P/(a + b). This needs a + b not to be zero.

    3. Not this one. You took a and b away, but each of them multiplies x. The sum ax + bx is x(a + b), and you divide by a + b. Go to M4.8.2. Try another.

    See the answer

    x = P/(a + b)

    Factorise: P = x(a + b). Divide both sides by (a + b): x = P/(a + b). This needs a + b not to be zero.

    • If you chose A: Dividing by a + b is not the same as dividing by a and by b. With P = 10, a = 2 and b = 3, x is 2, but P/a + P/b is more than 8. Go to M4.8.2.
    • If you chose C: You took a and b away, but each of them multiplies x. The sum ax + bx is x(a + b), and you divide by a + b. Go to M4.8.2.

  6. Make x the subject of ax = bx + c.

    Tap your answer.

    1. Not this one. You moved b without its x. The term is bx, so x is still in it. Take bx away from both sides. Go to M4.8.2. Try another.

    2. Not this one. Taking bx from both sides gives x(a − b) = c. The whole of a − b divides c, not only the a. Go to M4.8.2. Try another.

    3. Yes, that is right. Take bx from both sides: ax − bx = c. Factorise: x(a − b) = c. Divide by (a − b): x = c/(a − b), when a is not equal to b. Test it with a = 6, b = 2 and c = 4. Then 6x = 2x + 4 gives 4x = 4, so x = 1. And c/(a − b) = 4/4 = 1.

    See the answer

    x = c/(a − b)

    Take bx from both sides: ax − bx = c. Factorise: x(a − b) = c. Divide by (a − b): x = c/(a − b), when a is not equal to b. Test it with a = 6, b = 2 and c = 4. Then 6x = 2x + 4 gives 4x = 4, so x = 1. And c/(a − b) = 4/4 = 1.

    • If you chose A: You moved b without its x. The term is bx, so x is still in it. Take bx away from both sides. Go to M4.8.2.
    • If you chose B: Taking bx from both sides gives x(a − b) = c. The whole of a − b divides c, not only the a. Go to M4.8.2.

  7. Use t = (v − u)/a with v = 30, u = 6 and a = 4. What is t?

    See the answer

    t = 6

    Put the numbers in: t = (30 − 6)/4 = 24/4 = 6.

  8. Use x = (d − b)/(a − c) with a = 7, b = 1, c = 4 and d = 13. What is x?

    See the answer

    x = 4

    Put the numbers in: x = (13 − 1)/(7 − 4) = 12/3 = 4. Check in 7x + 1 = 4x + 13: 29 = 29.

  9. From an earlier chapter

    Solve 3(x + 4) = x + 18.

    See the answer

    x = 3

    Expand: 3x + 12 = x + 18. Take x from both sides: 2x + 12 = 18. Take 12 from both sides: 2x = 6. Divide by 2: x = 3.

Keep it

Copy this into your exercise book.

Undo the steps in reverse order, doing the same to both sides. If the subject is in two terms, collect them and factorise. Never divide by zero.

Draw: A machine with two rows: t, × a, at, + u, v, and the row back: v, − u, at, ÷ a, t.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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