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Cambridge Lower Secondary Mathematics Learners Book 8 2nd Edition Lynn Byrd, Greg Byrd, Chris Pearce - Solutions

An algebra chain is a sequence of expressions where an input number is substituted as the value of x in the first expression, and then the output of each expression is substituted as the value of x in the following expression.So, in the algebra chain below:3 is substituted for x in the expression
Write the inequality shown by this number line. Use the letter x. 05 5 10 15 20 25
Work out the value of x and y in this diagram.All measurements are in centimetres. 3+ +16 6(x+1) 3x+21 20
Factorise a 6x + 9 b 2y2 − 12y
Expand a x(x + 3)b 5y(7y − 4w)
a Use the formula K = mg to work out K when m = 12 and g = 4 .b Rearrange the formula K = mg to make m the subject.c Use your formula in part b to work out m when K = 75 and g = 10.
Jin thinks of a number, x.Write an expression for the number Jin gets when he divides the number by 2 then adds 5.
Zara is looking at this question.Draw a line linking each inequality on the left with: the correct smallest integer; the correct largest integer; and the correct list of integers. The first one has been done for you: a and ii and D and ZRead Zara’s comments.a What do you think of Zara’s method?
This is part of Sandeep’s classwork.a Sandeep has made two mistakes. What are they?b For each of these inequalities, write i the smallest integer that m could be ii the greatest integer that m could be ii a list of the integer values that m could be. Question 12 = m < 7 on on a number line. a
Samir combines two inequalities into one.a Use Samir’s method to combine each pair of inequalities into one.Use a number line to help if you want to.i m 8 ii m ⩽ 10 and m > 7 iii m > 0 and m 14 and m Give a reason for your answer.Discuss your answer with a partner. The two inequalities m
Write true (T) or false (F) for each statement.Part a has been done for you.a 7 ⩾ y > 3 means the same as 3 < y ⩽ 7 T b 15 > y ⩾ 5 means the same as 5 < y ⩽ 15 c 10 ⩾ y ⩾ −6 means the same as −6 ⩽ y ⩽ 10 d 8 > y ⩾ −8 means the same as −8 < y ⩽ 8
Arun and Sofia are looking at the inequality 2 In pairs or small groups discuss Arun’s and Sofia’s comments.What do you think? Explain your answers. I think the inequality 2 y>2 I think the inequality 2 y>9
For each of these inequalities, write i the smallest integer that y could be ii the largest integer that y could be iii a list of the integer values that y could be.a 3 < y < 8 b 4 < y ⩽ 7 c 0 ⩽ y < 6 d −10 ⩽ y ⩽ −6
This is part of Ryan’s homework.a Explain the mistakes Ryan has made and write the correct solutions.b Discuss your answers to part a with a partner.Make sure you have corrected all of Ryans’s mistakes. Question Use the inequality 12 x < 18 to write i the smallest integer that x could be the
Copy and complete these equivalent inequalities. 10 a C x > 8 is equivalent to 3x > y=7 is equivalent to y+3| b x < 3 is equivalent to x < 15 d y 2 is equivalent to y-4 |
Sofia and Zara are looking at the inequality x > 5.In pairs or small groups, discuss Sofia’s and Zara’s comments.a How can you show that Sofia and Zara are correct?b Write two different inequalities that are equivalent to x > 5.c Is it possible to say how many different inequalities there
Write the inequality that each of these number lines shows.Use the letter x. 14 15 16 17 b d 11 12 13 C -3-2 -1 0 1 -2 O 13 3 -2 st -5 6 8 10 12
Copy each number line and show each inequality on the number line. 50 a 4
2 Write each statement as an inequality. Part a has been done for you.a y is greater than or equal to 3 and less than 17 3< y
Write in words what each of these inequalities means.Part a has been done for you.a 6 < x < 11 x is greater than 6 and less than 11 b 12 ⩽ x ⩽ 18 c 0 < x ⩽ 20 d −9 ⩽ x < −1
a i Show the inequality 2 ⩽ x < 6 on a number line.ii List the possible integer values for.b i Show the inequality −5 < y ⩽ −1 on a number line.ii List the possible integer values for y.
Art has these cards.He chooses one blue card, the red card and one yellow card to make an equation.Which blue and yellow card should he choose to give him the equation with a the largest value for y b the smallest value for y?Explain your decisions and show that your answers are correct. 2y+14
13 This is part of Mo’s homework.You can see that instead of multiplying out the bracket, Mo’s first step is to divide both sides of the equation by 4.Use Mo’s method to solve these equations. 2b-3=-2b 2b=-2b+3 4b=3 3 b=4 Question Solve the equation 4(2b-3)=-8b Answer Divide both sides by
12 Solve these equations.Use the Tip box to help.a 5(2x + 3) + 2(x − 4) = 31 b 4(3x − 1) − 3(5 − 2x) = 35 c 2/3 y = 8 d 3/5 y + 1 = 19
The diagram shows the sizes of the two equal angles in an isosceles triangle.a Write an equation to represent the problem.b Solve your equation to find the value of x.c Work out the size of each of the angles in the triangle. 4x-6 2x + 18
The diagram shows the sizes of the angles in a triangle.a Write an equation to represent the problem.b Solve your equation to find the value of n.c Work out the size of each of the angles in the triangle. 2n+5 6n n-5
Work in a group of three or four. For each part of this question:i Write an equation to represent the problem.ii Compare the equation you have written with the equations written by the other members of your group. Decide who has written the correct equation in the easiest way.iii Solve the equation
Work out the value of x and y in each diagram. All measurements are in centimetres.Show how to check your answers are correct. 100 2y+15 8(y-1) 2(y+3) 3y+ 16 5x 3 37 5x-3 3x + 11 20 8y-4 b d 4y+5 16 3x+1 2(x+5) 25 X +17 4
Work out the value of y in each shape. All measurements are in centimetres.Show how to check your answers are correct. a 4(y-3) 2y+2 b 8y-5 C 2(y+6) 4(y-3) 3(y+5) ||
Work out the value of x in each isosceles triangle.All measurements are in centimetres. a 6x-3 C x + 35 5x 13 b 27 +20
Marcus and Sofia are discussing what equation to write to answer this question.The diagram shows an isosceles triangle.All measurements are in centimetres.Work out the value of y. 2y+7 5y-17
For each learner i write an equation to represent what they say ii solve your equation to find the value of x.The first one has been started for you. x-3=15 X -3+3 15+3 2 X 2 x x = x2 x = X
Copy and complete the workings to solve these equations. a 3x+5=26 (subtract 5 from both sides) (simplify) 3x+5-5-26-5 3xx = (divide both sides by 3) x= 3 (simplify) b 4(x-3)=24 (multiply out the brackets). x= 4x-12=24 (add 12 to both sides) 4x-12+12=24+12 (simplify) (divide both sides by 4)
Write if each of the following is a formula, an expression or an equation.a 3y + 7 = 35 b 6(x + 5)c T = 3a2 − 8d d 9u − vw
a Write if each of the following is a formula, an expression or an equation.i 4c + 3e ii P = 8h + b iii 9k − 2 = 16 b The diagram shows a rectangle.Work out the values of x and y. 5y-4cm 3(x+3)cm [3y+8cm 24 cm
Work with a partner.Take it in turns to define the following terms.a What is a factor?b What is the highest common factor?c What is factorising?How did your answers to a and b help with your answer to c?
13 The diagrams show two rectangles.The area of rectangle A is 2a2 + 18a.The perimeter of rectangle B is 14d − 10c.Write an expression for the length of each rectangle, in its simplest form. 2a A 3d B length length
Read what Zara says.Show that she is right.12 Read what Marcus says.Show that he is wrong.Explain the mistake Marcus has made. When I expand 5(2x+6)+2(3x-5), then collect like terms and finally factorise the result, I get the expression 4(4x+5)
Copy and complete these factorisations. a 2x+6y+8=2(x+3y+) 9xy+12y-15=3(3xy + -5) b 4y-8+4x=4(y- +x) d e 9y-y-xy=y(0-0) f 5x+2x+xy=x(5x++) 3y-9y+6xy=3y( -+)
Factorise each of these expressions.a 3x2 + x b 6y2 − 12y c 3b + 9b2 d 12n − 15n2 e 18y − 9x f 12y + 9x g 8xy − 4y h 15z + 10yz
Each expression on a yellow card has been factorised to give an expression on a blue card.Match each yellow card with the correct blue card. A 6x+12x B 6x+15x C 6x+9x D 6x+18x 3x(2x+5) ii 6x(x+3) iii 6x(x+2) iv 3x(2x+3)
In pairs or a small group, discuss what Zara and Sofia say.a What do you think? Explain your answer.b What is the highest common factor of i 8y and 4y2 ii 12p2 and 15p iii 4ab and 5a? I think the highest common factor of 6.x and 9.x is 3. I think the highest common factor of 6.x and 9.x is 3x.
Factorise each of these expressions.Make sure you use the highest common factor.a 10z + 5 b 8a − 4 c 14 + 21x d 18 − 24z
Factorise each of these expressions.Each one has a highest common factor of 3.a 18 + 21p b 3y − 18 c 9 + 15m d 12 − 27x
Factorise each of these expressions.Each one has a highest common factor of 2.a 2x + 4 b 4b − 6 c 8 + 10y d 18 − 20m
In pairs or a small group, discuss what Marcus and Arun say.Do you agree with Arun? Explain your answer. When I factorise 6x+ 18 I get 3(2x+6) I don't, I get 6(x+3) so one of us must be wrong!
Copy and complete these factorisations.All the numbers you need are in the cloud. 1 2 5 7
Copy and complete these factorisations.All the numbers you need are in the cloud. a 3x+15=3(x+ C 14-28x=7-4x) b 10y-15=5(2y- d 12-9y=3(-3y)
Factorise these expressions.a 2x + 10 b 8 − 12y c 4a + 8ab d x2 − 5x
Work with a partner to answer this question.Here are six expressions.A x(5x + 2) + 3x(4x + 1) B y(y2 + 4) + 6y2(y + 8)C 7p(2p2 + 7p −1) + 9p D 6k + 18 − 3k(4 − 5k2 )E 5n(n2 − 4) − 2n2(n + 2) F 8m ( m + 3) − 2m (4m − 3)a Choose one of the expressions and ask your partner to expand the
This is part of Shen’s homework. He has made a mistake in every question.a Explain what Shen has done wrong.b Work out the correct answers. Question Expand and simplify 1 8(x+5)-3(2x+7) 3 2y (y+5x)+x(3x+4y) Answers 2 a(2b+c)+b(3c-2a) 18(x+5)-3(2x+7)=8x+40-6x+21=2x+61 2 a(2b+c)+b(3c-2a) = 2ab+ ac
Expand each expression and simplify by collecting like terms.a x(x + 2) + x(x + 5) b z(2z + 1) + z(4z + 5)c u(2u + 5) − u(u + 3) d w(6w + 2x) − 2w(2w − 9x)
Work with a partner to discuss this question.Look at this expansion. x(2x + 5) + 3x(2x + 4) = 2x2 + 5x + 6x2 + 12x a How would the expansion change if the + changed to −?Here is the expansion again. x(2x + 5) + 3x(2x + 4) = 2x2 + 5x + 6x2 + 12x b How would the expansion change if both the +
Here are some expression cards.Sort the cards into groups of equivalent expressions. A 2x(8x + 6x) B 10x(3x+2) C 2x (12x+9) D 2x (10+15x2) E 4x (4x+3) F 3x (6+8x) G 5(6x+4x) H 6x (3x+4x) I x (12+16x)
Jing, Jun and Amira compare the methods they use to expand the bracket 5k(6m − 8k).a What do you think about Jing, Jun and Amira’s methods?b Which method do you think is best for expanding brackets correctly? Explain why.c Use your favourite method to expand i 2x(x + 3y) ii 3y(5y + 6)iii 4b(6b
In pairs or a small group, discuss what Zara and Arun say.Do you agree with Arun? Explain your answer. When I expand 2d(4c-7a), I get 8cd-14ad I don't, I get 8dc-14da so one of us must be wrong!
Expand each expression.a y(y + 8)b z(2w − 1)c m(m − 4)d n(2n + 5)e n(9 − 8n)f a(1 − 3b)g e(2e + 7f)h g(3h + 7g)i h(2h − 5k) j d(3c − 5e)
Copy and complete the working.Expand the brackets. a x(y+3)=xxy+xx3 = xy+| p(3+4p) px3+px 4p =+4p b d y(y-2)=yxy-yx2 = 12 - q(6q-15)=qx6q-qx15 = -0
Expand each expression.a 4(x + 6)b 7(z − 2)c 2(a + 8)d 6(3 − 4e)e 2(2p + 3q)f 9(6t − 2s)g 7(6xy − 2z)h 5(2x + y + 4)
1 Copy and complete the working.Expand the brackets. a C 3(x+4)=3xx+34 =3x+ 9(3q-4)=9x3q-9x4 = 0-0 b 8(y-2)=8xy-82 =8y-
a Expand these expressions. i 3(2b + 5) ii a(a − 3)b Expand and simplify this expression. 4(2x + 3x2) − x(6 + x)
Polly and Theo use different methods to work out the answer to a question.This is what they write.a Look at Polly and Theo’s methods.Do you understand both methods?Do you think you would be able to use both methods?b Which method do you prefer and why?c Use your preferred method to answer these
a Use the formula f wp = to work out the value of f when w = 60 and p = 12.b Rearrange the formula f w p = to make w the subject.c Use your formula to work out the value of w when f = 0.25 and p = 52.
a Use the formula h = k − d to work out the value of h when k = 72 and d = 37.b Rearrange the formula h = k − d to make k the subject.c Use your formula to work out the value of k when h = 0.42 and d = 1.83
a Use the formula T = mg to work out the value of T when m = 4.5 and g = 10.b Rearrange the formula T = mg to make m the subject.c Use your formula to work out the value of m when T = 320 and g = 10.
Xavier uses this formula to work out the volume of a triangular prism.V = bhl/2 where: V is the volume; b is the base; h is the height; l is the length.Xavier compares two prisms.Prism A has a base of 8 cm, height of 5 cm and length of 18 cm.Prism B has a base of 9 cm, height of 14 cm and length of
Use the formula C = πd to a estimate the value of C when d = 19 m b calculate the value of C when d = 19 m.Give your answer to one decimal place.
10 The height of a horse is measured in hands (H) and inches (I).This formula is used to work out the height of a horse in centimetres (C ).C = 2.5(4H + I) where: C is the number of centimetres H is the number of hands I is the number of inches.Sasha has a horse with a height of 16 hands and 1
9 Use the formula F = ma to work out F when a m = 6 and a = 2 b m = 18 and a = 3 c m = 8 and a = −4.
8 Use the formula v = u + 10t to work out the value of v when a u = 5 and t = 12 b u = 8 and t = 15 c u = 0 and t = 20.
7 This is how a taxi company works out the cost of a journey for a customer:There is a fixed charge of $6 plus $2 per kilometre.a Write a formula for the cost of a journey, in i words ii letters.b Use your formula in part aii to work out the cost of a journey of 35 km.
a Write a formula for the number of months in any number of years, in i words ii letters.b Use your formula in part aii to work out the number of months in 8 years.
This is part of Oditi’s homework. She has made a mistake in her working.a Explain the mistake she has made.b Work out the correct answer.c Work out the value of 2y3 when y = −3. =-2. Question Work out the value of 5x when x = Answer 5x3 = 5x (-2) = (-10) =-10x-10x-10 = -1000
This is part of Dakarai’s homework. He has made a mistake in his working.a Explain the mistake he has made.b Work out the correct answer.c Work out the value of y2 + 4 when y = −5. Question Work out the value of x-8 when x=-3. Answer x-8=(-3)2-8 =-3x-3-8 =-9-8 =-17
Work out the value of each expression. a x+5 when x=4 b 10-y when y=5 C g2+h when g=3 and h=6 d m-n when m=7 and n=8 e 4k when k=2 f 3r when r=1 g 2y when y = 3 h x-5 when x=2 i 20-w when w=4 jwhen y=4
Work out the value of each expression.a 8m − 5 when m = −2 b 3z + v when z = 8 and v = −20 c 2x + 3y when x = 4 and y = 5 d 20 − 3n when n = 9 e u/2 − 5 when u = 4 f p/5 + q/2 when p = 30 and q = −8
Copy and complete the working to find the value of each expression. a p+5 when p=-3 b q-6 when q=4 6h when h=-3 d when j=-20 e f 4 a+b when a 6 and b=-3 c-d when c=25 and d=32 p+5=-3+5=[ 9-6=4-6= 6h=6x-3= j-20 4 a+b=6+-3=6-3= c-d=25-32=
a Work out the value of the expression 2x + 4y when x = 5 and y = −2.b Work out the value of the expression 3x2 + 4 when x = 10.c Write a formula for the number of hours (h) in any number of days (d), using i words ii letters.d Use the formula in part c to work out the number of hours in 7 days.e
15 Brad thinks of a number, y.Choose the correct expression from the cloud for when Brad a adds 5 to one-half of y, then multiplies by 6 b adds 6 to one-fifth of y, then multiplies by 2 c adds 2 to five-sixths of y, then multiplies by 6 d adds 5 to two-fifths of y, then multiplies by 6.Which
14 The price of one kilogram of apples is $a.The price of one kilogram of bananas is $b.The price of one kilogram of carrots is $c.Write an expression for the total cost of a one kilogram of apples and half a kilogram of bananas b two kilograms of bananas and three-quarters of a kilogram of carrots
13 The price of one bag of cement is $c.The price of one bag of gravel is $g.The price of one bag of sand is $s.Write an expression for the total cost of a one bag of cement and three bags of sand b three bags of cement, four bags of gravel and six bags of sand.
12 The shortest side of a triangle is y cm.The second side is 3 cm longer than the shortest side.The third side is twice as long as the second side.Write an expression, in its simplest form, for the perimeter of the triangle.
Write an expression for a the perimeter of this rectangle b the area of this rectangle. 4bcm acm
11 This is part of Maya’s homework.Use Maya’s method to write an expression for the perimeter and area of each of these rectangles. Simplify each expression. a 6bcm acm b 7ccm dcm
10 a Write an expression for each description.i one-half of x add 8 ii three-quarters of x subtract 12 iii 7 add four-fifths of x iv 20 subtract five-ninths of x b Describe each expression in words. +2 ii 5x-4 7x iv 3+ 3 8 iii 8-2x !!!
9 This is part of Pedro’s classwork.Are Pedro’s answers correct? If not, write the correct answers for him.Write an expression for these.a one-third of x add 4 b 5 subtract two-fifths of y
8 a Sort these cards into groups of equivalent expressions.b Which card is in a group on its own? A 3xx 4 x+3 B C 4 4 x x D Exx 3 34 FL G+ 3.x 4x 3+x +X H I J 4 3 4
7 In pairs or in a small group, discuss.Sofia, Zara and Arun discuss what to write for this problem.‘I think of a number, n. I divide by 3, then multiply by 2.’What do you think?Make a conjecture and convince the other members of your group. I think the expression 2n is I think the expression
6 Kia thinks of a number, x.Write an expression for the number Kia gets when she:a divides the number by 3, then adds 1 b adds 1 to the number, then divides by 3 c subtracts 1 from the number, then divides by 3 d divides the number by 3, then subtracts 1.
5 In pairs or in a small group, discuss.Sofia and Zara discuss what to write for this problem.‘I think of a number, n. I halve the number then add 4.’What do you think?Make a conjecture and convince the other members of your group. I think the expression is +4 2 I think the expression is n+4 2
4 Match each description with the correct expression.The first one has been done for you: a and iv.a Multiply n by 5 and subtract 4 b Add 4 and n, then multiply by 5 c Multiply n by 5 and add 4 d Add 5 and n, then multiply by 4 e Subtract 4 from n, then multiply by 5 f Subtract 5 from n, then
3 a Jake thinks of a number, n.Write an expression for the number Jake gets when he:i multiplies the number by 6, then adds 1 ii divides the number by 4, then adds 5 iii multiplies the number by 2, then subtracts 3 iv divides the number by 10, then subtracts 7.b Jake thinks of the number 20. Work
2 a Tanesha has a box that contains x DVDs.Choose the correct expression from the cloud that shows the total number of DVDs she has in the box when i she takes 2 out ii she puts in 2 more iii she takes out half of the DVDs iv she doubles the number of DVDs in the box.b Tanesha starts with 12 DVDs
1 Copy and complete these sentences. Use the words from the cloud.In the ................... 4x + 9, x is a ....................4x and 9 are ................... of the expression.4 is the ................... of x. 9 is a ....................The expression is not equal to anything so cannot be
Tyler thinks of a number, x. Write an expression for the number Tyler gets when he a doubles the number and subtracts 3 b divides the number by 3 and adds 2 c adds 2 to the number, then multiplies by 4.
6 Write the inequality shown by this number line. 2 3 4 5 6
5 Solve these equations.a n + 12 = 15 b m − 7 = 2 c 3p = 27 d 2r + 7 = 19
4 Expand the brackets.a 4(x + 3) b 6(2 − 3y)
3 Simplify these expressions by collecting like terms.a 3c + 4c + 9d − 2d b 4xy + 7yz − 2xy + zy
2 Work out the value of p − q when p = 15 and q = 3

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