Chapter 8 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

M4.8.8 Chapter 8

Word problems and regions

A yard has hens and goats: 30 heads and 84 legs. How many of each?

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

  • Write two equations from a word problem and solve them
  • Shade the region on a graph that fits two inequalities

A word problem gives facts about two unknown numbers. Turn each fact into an equation. The two equations together are simultaneous equations, and you can solve them with the methods of this book.

Take the yard. Let x be the number of hens and y the number of goats. Every animal has one head, so x + y = 30. A hen has 2 legs and a goat has 4, so 2x + 4y = 84.

Draw it first

Facts into equations

  1. Two equations labelled (1) and (2). Step 1 writes x + y = 30, with the note heads. Step 2 writes 2x + 4y = 84, with the note legs.x+y=30(1)heads

    1 Count heads. One head each: x + y = 30.

  2. Two equations labelled (1) and (2). Step 1 writes x + y = 30, with the note heads. Step 2 writes 2x + 4y = 84, with the note legs.x+y=30(1)headsx+y=30(1)heads2x+4y=84(2)legs

    2 Count legs. 2 for each hen and 4 for each goat: 2x + 4y = 84.

Worked example

A yard has hens and goats: 30 heads and 84 legs. Hens have 2 legs and goats have 4. How many of each?

  1. Let x be the number of hens and y the number of goats.
  2. Heads: x + y = 30. Legs: 2x + 4y = 84.
  3. Multiply the first equation by 2: 2x + 2y = 60. Subtract it from the second: 2y = 24.
  4. Divide by 2: y = 12. Then x = 30 − 12 = 18.
  5. Check the legs: 2 × 18 + 4 × 12 = 36 + 48 = 84.

Answer: 18 hens and 12 goats

Always check against the words, not only the equations. 18 + 12 = 30 heads is right, and 84 legs is right. Make sure the answers make sense: a number of animals is a whole number and is not negative.

Now a shape. A rectangle has a perimeter of 34 cm. Its length is 5 cm more than its width. Let l be the length and w the width, in cm.

A rectangle with its length l written above it and its width w written on its left side.lw
The perimeter is 2l + 2w. The length l is w + 5.

Worked example

A rectangle has a perimeter of 34 cm. Its length is 5 cm more than its width. Find the sides.

  1. Let l be the length and w the width, both in cm.
  2. The perimeter gives 2l + 2w = 34. The length gives l = w + 5.
  3. Put w + 5 in place of l: 2(w + 5) + 2w = 34, so 4w + 10 = 34.
  4. Take 10 from both sides: 4w = 24, so w = 6.
  5. Then l = 6 + 5 = 11.
  6. Check: 2 × 11 + 2 × 6 = 34 cm.

Answer: The width is 6 cm and the length is 11 cm.

Now inequalities with two letters. The inequality y > x is true for many points, not for one. The points that make it true fill a region of the graph. The line y = x is the boundary.

Draw the boundary solid when the sign is ≤ or ≥, because points on the line fit. Draw it dashed when the sign is < or >, because points on the line do not fit. Then test a point to find which side to hatch. Hatching means shading with lines, so a question that says shade the region wants the same thing.

Draw it first

Hatch a region

  1. A graph with the x axis x and the y axis y. Step 1 draws three boundary lines: a solid line named x = 1, a dashed line named y = x, and a solid line named y = 6. Step 2 marks a test point T. Step 3 hatches the triangle that fits x ≥ 1, y > x and y ≤ 6.x = 1y = xy = 61234567123456780xy

    1 Draw each boundary. x = 1 and y = 6 are solid. y = x is dashed, because the sign is strict.

  2. A graph with the x axis x and the y axis y. Step 1 draws three boundary lines: a solid line named x = 1, a dashed line named y = x, and a solid line named y = 6. Step 2 marks a test point T. Step 3 hatches the triangle that fits x ≥ 1, y > x and y ≤ 6.Tx = 1y = xy = 61234567123456780xy

    2 Choose a point that is on none of the lines, such as T (2, 4), and test it in each inequality.

  3. A graph with the x axis x and the y axis y. Step 1 draws three boundary lines: a solid line named x = 1, a dashed line named y = x, and a solid line named y = 6. Step 2 marks a test point T. Step 3 hatches the triangle that fits x ≥ 1, y > x and y ≤ 6.Tx = 1y = xy = 61234567123456780xy

    3 Hatch the region that fits all three. T fits each one: 2 ≥ 1, 4 > 2 and 4 ≤ 6.

To find the side of one line, test a point that is not on it. For y > x + 1, try (0, 0). It gives 0 > 1, which is false, so the origin is on the wrong side. Hatch the other side, above the line.

Worked example

Show the region that fits x ≥ 1, y ≥ 2 and x + y ≤ 8.

  1. Draw the solid lines x = 1, y = 2 and x + y = 8. They are solid because the signs are ≥ and ≤.
  2. Test (0, 0) in x + y ≤ 8: 0 + 0 = 0, and 0 ≤ 8 is true. So hatch the side of x + y = 8 that holds the origin.
  3. Keep only the part with x ≥ 1 and y ≥ 2. The smallest y in the region is 2.
  4. The corners are where pairs of lines meet. Where y = 2 meets x + y = 8, x + 2 = 8, so x = 6.
  5. Where x = 1 meets x + y = 8, 1 + y = 8, so y = 7.
  6. The region is a triangle with corners (1, 2), (6, 2) and (1, 7).

Answer: The triangle with corners (1, 2), (6, 2) and (1, 7).

A graph with the x axis x and the y axis y. A solid line named x = 1, a solid line named y = 2 and a solid line named x + y = 8 make a hatched triangle with corners at (1, 2), (6, 2) and (1, 7).x = 1y = 2x + y = 812345678123456780xy
The hatched triangle fits all three. Its edges are solid, so points on the edges count.

See it three ways

With things

Amina takes at most 6 pieces of fruit: x apples and y mangoes. She wants at least 1 apple and at least 1 mango.

As a picture

A graph with the x axis apples x and the y axis mangoes y. A solid line named x = 1, a solid line named y = 1 and a solid line named x + y = 6 make a hatched triangle.x = 1y = 1x + y = 6123456712345670apples xmangoes y

In words

Every whole-number point inside the triangle, or on its edge, is a basket Amina can take. For example, 2 apples and 3 mangoes is 5 pieces.

In symbols

x ≥ 1, y ≥ 1 and x + y ≤ 6.

Try it

Hatch your own region

You need: squared paper, a ruler, a pencil.

  1. Draw the lines x = 1 and y = 1 solid, and x + y = 7 dashed.
  2. Test the point (2, 2) in all three signs: x ≥ 1, y ≥ 1 and x + y < 7.
  3. Hatch the triangle that fits all three.

What you will see: The point (2, 2) fits, so it is inside. The dashed edge is not included. So (3, 4), where x + y = 7, is not in the region.

If you cannot try it, this is what happens.
A graph with the x axis x and the y axis y. A solid line named x = 1, a solid line named y = 1 and a dashed line named x + y = 7 make a hatched triangle.x = 1y = 1x + y = 7123456712345670xy

A wrong idea many people have

You can pick either side of the boundary to hatch without testing.

Why it feels right: The sign y > x + 2 sounds like "up", so it feels as if the side is obvious.

The test that breaks it: Test (0, 0) in y > x + 2. It gives 0 > 2, which is false. So the side with the origin is not the region. Test (0, 5): 5 > 2 is true.

What is true: Put a point that is not on the line into the inequality. If it is true, hatch its side. If it is false, hatch the other side.

A wrong idea many people have

A boundary line is always drawn solid.

Why it feels right: Most of the lines in a graph are solid, so it feels like the normal way.

The test that breaks it: Take y < x. The point (3, 3) is on the line y = x. Put it in: 3 < 3 is false. So (3, 3) is not in the region, and the line must show that it is left out.

What is true: A solid line says the points on it are included (≤ or ≥). A dashed line says they are not (< or >).

Key words

region
the part of a graph where every point fits the inequalities.
boundary
the line that edges a region. It is solid when included and dashed when not.

Back to your guess

The answer is A: 18 hens and 12 goats.

Hens have 2 legs and goats have 4. 18 hens and 12 goats give 36 + 48 = 84 legs and 30 heads. The other pairs have 30 heads, but the legs are 96 and 90.

Check yourself

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

  1. A farmer has chickens (2 legs each) and cows (4 legs each): 20 heads and 56 legs. Let c be the number of chickens and w the number of cows. Which equations fit?

    Tap your answer.

    1. Not this one. You swapped the totals. Heads add to 20 and legs add to 56. Go to M4.8.8. Try another.

    2. Not this one. You gave 4 legs to the chickens and 2 to the cows. Chickens have 2 legs each, so the chicken term is 2c. Go to M4.8.8. Try another.

    3. Not this one. Legs are added, not multiplied. Each chicken adds 2 and each cow adds 4, so the legs equation is 2c + 4w. Go to M4.8.8. Try another.

    4. Yes, that is right. One head each gives c + w = 20. Legs give 2c + 4w = 56. Here c is the number of chickens and w the number of cows.

    See the answer

    c + w = 20 and 2c + 4w = 56

    One head each gives c + w = 20. Legs give 2c + 4w = 56. Here c is the number of chickens and w the number of cows.

    • If you chose A: You swapped the totals. Heads add to 20 and legs add to 56. Go to M4.8.8.
    • If you chose B: You gave 4 legs to the chickens and 2 to the cows. Chickens have 2 legs each, so the chicken term is 2c. Go to M4.8.8.
    • If you chose C: Legs are added, not multiplied. Each chicken adds 2 and each cow adds 4, so the legs equation is 2c + 4w. Go to M4.8.8.

  2. Solve the farmer's equations c + w = 20 and 2c + 4w = 56. Here c is the number of chickens and w the number of cows. How many of each are there?

    Tap your answer.

    1. Not this one. Check the legs: 2 × 8 + 4 × 12 = 64, not 56. The numbers are the wrong way round. Go to M4.8.8. Try another.

    2. Yes, that is right. Multiply the first equation by 2: 2c + 2w = 40. Subtract it from the second: 2w = 16, so w = 8. Then c = 12. Legs: 24 + 32 = 56.

    3. Not this one. This has 20 heads, but the legs are 2 × 10 + 4 × 10 = 60, not 56. Go to M4.8.8. Try another.

    4. Not this one. This has 20 heads, but the legs are 2 × 14 + 4 × 6 = 52, not 56. Go to M4.8.8. Try another.

    See the answer

    12 chickens and 8 cows

    Multiply the first equation by 2: 2c + 2w = 40. Subtract it from the second: 2w = 16, so w = 8. Then c = 12. Legs: 24 + 32 = 56.

    • If you chose A: Check the legs: 2 × 8 + 4 × 12 = 64, not 56. The numbers are the wrong way round. Go to M4.8.8.
    • If you chose C: This has 20 heads, but the legs are 2 × 10 + 4 × 10 = 60, not 56. Go to M4.8.8.
    • If you chose D: This has 20 heads, but the legs are 2 × 14 + 4 × 6 = 52, not 56. Go to M4.8.8.

  3. A rectangle has a perimeter of 30 cm. Its length is twice its width. What is the width?

    Tap your answer.

    1. Yes, that is right. The length is l = 2w. Put it in 2l + 2w = 30: 4w + 2w = 30, so 6w = 30 and w = 5. The length is 10 cm.

    2. Not this one. That is the length, which is twice the width. The width is the shorter side. Go to M4.8.8. Try another.

    3. Not this one. That is half the perimeter, l + w. The width is only part of that. Go to M4.8.8. Try another.

    4. Not this one. You divided the perimeter by 4, as for a square. The sides here are different, so 4 equal sides do not fit. Go to M4.8.8. Try another.

    See the answer

    5 cm

    The length is l = 2w. Put it in 2l + 2w = 30: 4w + 2w = 30, so 6w = 30 and w = 5. The length is 10 cm.

    • If you chose B: That is the length, which is twice the width. The width is the shorter side. Go to M4.8.8.
    • If you chose C: That is half the perimeter, l + w. The width is only part of that. Go to M4.8.8.
    • If you chose D: You divided the perimeter by 4, as for a square. The sides here are different, so 4 equal sides do not fit. Go to M4.8.8.

  4. Which inequalities describe the hatched region?

    A graph with the x axis x and the y axis y. A solid line named x = 2, a solid line named y = 1 and a solid line named x + y = 7 make a hatched triangle.x = 2y = 1x + y = 71234567123456780xy
    Tap your answer.

    1. Not this one. The lines are solid, so the edges are included. That needs ≥ and ≤, not > and <. Go to M4.8.8. Try another.

    2. Not this one. The hatching is to the right of x = 2 and above y = 1, and below x + y = 7. All three signs are the wrong way. Go to M4.8.8. Try another.

    3. Yes, that is right. The region is on and right of x = 2, on and above y = 1, and on and below x + y = 7. The lines are solid, so the signs include equals.

    4. Not this one. The hatching is below the line x + y = 7. Test (3, 2): 3 + 2 = 5, which is not 7 or more. Go to M4.8.8. Try another.

    See the answer

    x ≥ 2, y ≥ 1 and x + y ≤ 7

    The region is on and right of x = 2, on and above y = 1, and on and below x + y = 7. The lines are solid, so the signs include equals.

    • If you chose A: The lines are solid, so the edges are included. That needs ≥ and ≤, not > and <. Go to M4.8.8.
    • If you chose B: The hatching is to the right of x = 2 and above y = 1, and below x + y = 7. All three signs are the wrong way. Go to M4.8.8.
    • If you chose D: The hatching is below the line x + y = 7. Test (3, 2): 3 + 2 = 5, which is not 7 or more. Go to M4.8.8.

  5. Why is the line for y > x + 1 drawn dashed?

    Tap your answer.

    1. Not this one. Dashed or solid says nothing about the side. Testing a point decides the side. Go to M4.8.8. Try another.

    2. Yes, that is right. A point on the line makes y equal to x + 1, and y > x + 1 needs y to be more. So no point on the line fits, and the line is dashed.

    3. Not this one. The size of the region does not matter. Only the sign matters: strict signs give a dashed line. Go to M4.8.8. Try another.

    See the answer

    The sign is strict, so points on the line do not fit.

    A point on the line makes y equal to x + 1, and y > x + 1 needs y to be more. So no point on the line fits, and the line is dashed.

    • If you chose A: Dashed or solid says nothing about the side. Testing a point decides the side. Go to M4.8.8.
    • If you chose C: The size of the region does not matter. Only the sign matters: strict signs give a dashed line. Go to M4.8.8.

  6. The region fits y ≥ 1, y < x and x ≤ 5. Which of the four points is inside it?

    A graph with the x axis x and the y axis y. A solid line named y = 1, a dashed line named y = x and a solid line named x = 5 make a hatched triangle. Four points are named A at (3, 3), B at (6, 2), C at (2, 4) and D at (4, 2).ABCDy = 1y = xx = 5123456712345670xy
    Tap your answer.

    1. Not this one. This point is on the dashed line y = x, and y < x does not include it. Points on a dashed line are not in the region. Go to M4.8.8. Try another.

    2. Not this one. The line x = 5 is solid, but 6 is beyond it. The region needs x to be 5 or less. Go to M4.8.8. Try another.

    3. Not this one. Here y is 4 and x is 2, so y > x. The region needs y < x, below the dashed line. Go to M4.8.8. Try another.

    4. Yes, that is right. For D (4, 2): 2 ≥ 1, 2 < 4 and 4 ≤ 5 are all true. The other points each break one of the rules.

    See the answer

    D (4, 2)

    For D (4, 2): 2 ≥ 1, 2 < 4 and 4 ≤ 5 are all true. The other points each break one of the rules.

    • If you chose A: This point is on the dashed line y = x, and y < x does not include it. Points on a dashed line are not in the region. Go to M4.8.8.
    • If you chose B: The line x = 5 is solid, but 6 is beyond it. The region needs x to be 5 or less. Go to M4.8.8.
    • If you chose C: Here y is 4 and x is 2, so y > x. The region needs y < x, below the dashed line. Go to M4.8.8.

  7. A shed holds bicycles (2 wheels) and tricycles (3 wheels), 14 in all, with 33 wheels. How many tricycles are there?

    See the answer

    5 tricycles

    Let b be the number of bicycles and t the number of tricycles. Then b + t = 14 and 2b + 3t = 33. Multiply the first by 2: 2b + 2t = 28. Subtract: t = 5.

  8. A rectangle has a perimeter of 40 cm. Its length is 4 cm more than its width. What is the width?

    See the answer

    8 cm

    Let the width be w. Then the length is w + 4 and 2(w + 4) + 2w = 40. So 4w + 8 = 40, 4w = 32 and w = 8.

  9. From an earlier chapter

    The lines y = x + 3 and y = 9 − x cross. Find the crossing point. Give x first, then y.

    See the answer

    x = 3, y = 6

    Put the right sides equal: x + 3 = 9 − x, so 2x = 6 and x = 3. Then y = 3 + 3 = 6.

Keep it

Copy this into your exercise book.

Name the unknowns, write a fact as each equation, solve, check in the words. For a region: solid for ≤ and ≥, dashed for < and >, and test a point.

Draw: A triangle made of three lines, hatched inside, with one test point marked.

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

No internet tomorrow? Save the whole book as one file (989 KB). It opens on any phone.