Fluency
- WE8 Write linear equations for each of the following statements, using x to represent the unknown.
(Do not attempt to solve the equations.)
a. When 3 is added to a certain number, the answer is 5.
b. Subtracting 9 from a certain number gives a result of 7.
c. Seven times a certain number is 24.
d. A certain number divided by 5 gives a result of 11.
e. Dividing a certain number by 2 equals −9.
f. Three subtracted from five times a certain number gives a result of −7.
g. When a certain number is subtracted from 14 and this result is then multiplied by 2, the result is −3.
h. When 5 is added to three times a certain number, the answer is 8.
i. When 12 is subtracted from two times a certain number, the result is 15.
j. The sum of 3 times a certain number and 4 is divided by 2, which gives a result of 5.
- MC Which equation matches the following statement?
a. A certain number, when divided by 2, gives a result of −12.
a. x = −12
2
b. 2x = −12 c. x
2 = −12 d. x
12 = −2
b. Dividing 7 times a certain number by −4 equals 9.
a. x
−4 = 9 b. −4x
7 = 9 c. 7 + x
−4 = 9 d. 7x
−4 = 9
c. Subtracting twice a certain number from 8 gives 12.
a. 2x − 8 = 12 b. 8 − 2x = 12 c. 2 − 8x = 12 d. 8 −(x + 2) = 12
d. When 15 is added to a quarter of a number, the answer is 10.
a. 15 + 4x = 10 b. 10 = x
4 + 15 c. x +15
4 = 10 d. 15 +
4
x = 10
Understanding
When a certain number is added to 3 and the result is multiplied by 4, the answer is the same as when
the same number is added to 4 and the result is multiplied by 3. Find the number.
WE9 John is three times as old as his son Jack, and the sum of their ages is 48. How old is John?
In one afternoon’s shopping Seedevi spent half as much money as Georgia, but $6 more than Amy. If
the three of them spent a total of $258, how much did Seedevi spend?
These rectangular blocks of land have the same area. Find the dimensions of each block, and the area.
x + 5
20
x
30
Reasoning
- A square pool is surrounded by a paved area that is 2 metres wide. If the area of the paving is 72m2
,
what is the length of the pool?
2 m
Maria is paid $11.50 per hour, plus $7 for each jacket that she sews. If she earned $176 for one 8-hour
shift, how many jackets did she sew?
Mai hired a car for a fee of $120 plus $30 per day. Casey’s rate for his car hire was $180 plus
$26 per day. If their final cost and rental period were the same, how long was the rental period?
WE10 The cost of producing music CDs is quoted as $1200 plus $0.95 per disk. If Maya’s recording
studio has a budget of $2100, how many CDs can she have made?
Joseph wishes to have some flyers delivered for his grocery business. Post Quick quotes a price of
$200 plus 50 cents per flyer, while Fast Box quotes $100 plus 80 cents per flyer.
a. If Joseph needs to order 1000 flyers, which distributor would be cheaper to use?
b. For what number of fliers will the cost be the same for either distributor?
Problem solving
- A number is multiplied by 8 and 16 is then subtracted. The result is the same as 4 times the original
number minus 8. What is the number?
- Carmel sells three different types of healthy drinks; herbal,
vegetable and citrus fizz. One hour she sells 4 herbal,
3 vegetable and 6 citrus fizz for $60.50. The next hour she sells
2 herbal, 4 vegetable and 3 citrus fizz. The third hour she sells
1 herbal, 2 vegetable and 4 citrus fizz. The total amount in cash
sales for the three hours is $136.50. Carmel made $7 less in the
third hour than she did in the second hour of sales.
Determine her sales in the fourth hour, if Carmel sells
2 herbal, 3 vegetable and 4 citrus fizz.
- A rectangular swimming pool is surrounded by a path which is
enclosed by a pool fence. All measurements are in metres and are not to scale
in the diagram shown.
a. Write an expression for the area of the entire fenced-off section.
b. Write an expression for the area of the path surrounding the pool.
c. If the area of the path surrounding the pool is 34 m2
, find the dimensions of
the swimming pool.
d. What fraction of the fenced-off area is taken up by the pool?
Reflection
Why is it important to define the pronumeral used when forming a linear equation to solve a
problem?
4.6 Rearranging formulas
4.6.1 Rearranging (transposing) formulas
• Formulas are generally written in terms of two or more pronumerals or variables.
• One pronumeral is usually written on one side of the equal sign.
• When rearranging formulas, use the same methods as for solving linear equations (use inverse
operations in reverse order).
The difference between rearranging formulas and solving linear equations is that rearranging
formulas does not require a value for the pronumeral(s) to be found.
• The subject of the formula is the pronumeral or variable that is written by itself. It is usually written
on the left-hand side of the equation.
• A formula is simply an equation that is used for some specific purpose. By now you will be familiar
with many mathematical or scientific formulas.
For example, C = 2πr relates the circumference of a circle to its radius. When the formula is
shown in this order, C is called the subject of the formula. The formula can be transposed
(rearranged) to make r the subject.
C = 2πr Divide both sides by2π.
C
2π = 2πr
2π
C
2π = r
or r = C
2π
Nowr is the subject.
Fence
2 5
x + 2
x + 4
WORKED EXAMPLE 11 TI | CASIO
Rearrange each formula to make x the subject.
a y = kx + m b 6(y + 1) = 7(x − 2)
THINK WRITE
a 1 Subtract m from both sides. a y = kx + m
y − m = kx
2 Divide both sides by k. y − m
k = kx
k
y − m
k = x
3 Rewrite the equation so that x is on the
left-hand side.
x = y − m
k
b 1 Expand the brackets. b 6(y + 1) = 7(x − 2)
6y + 6 = 7x − 14
2 Add 14 to both sides. 6y + 20 = 7x
3 Divide both sides by 7. 6y + 20
7 = x
4 Rewrite the equation so that x is on the left-hand side. x = 6y + 20
7
WORKED EXAMPLE 12
For each of the following make the variable shown in brackets the subject of the formula.
a g = 6d − 3 (d) b a = v − u
t (v)
THINK WRITE
a 1 Add 3 to both sides. a g = 6d − 3
g + 3 = 6d
2 Divide both sides by 6. g + 3
6 = d
3 Rewrite the equation so that d is on the
left-hand side. d = g + 3
6
b 1 Multiply both sides by t. b a = v − u
t
at = v − u
2 Add u to both sides. at + u = v
3 Rewrite the equation so that v is on the
left-hand side.
v = at + u
Exercise 4.6 Rearranging formulas
Fluency
- WE11 Rearrange each formula to make x the subject.
a. y = ax b. y = ax + b c. y = 2ax − b
d. y + 4 = 2x − 3 e. 6(y + 2) = 5(4 − x) f. x(y − 2) = 1
g. x(y − 2) = y + 1 h. 5x − 4y = 1 i. 6(x + 2) = 5(x − y)
j. 7(x − a) = 6x + 5a k. 5(a − 2x) = 9(x + 1) l. 8(9x − 2) + 3 = 7(2a − 3x)
- WE12 For each of the following, make the variable shown in brackets the subject of the formula.
a. g = 4P − 3 (P) b. f = 9c
5 (c) c. f = 9c
5
- 32 (c)
d. V = IR (I) e. v = u + at (t) f. d = b2 − 4ac (c)
g. m = y − k
h (y) h. m = y − a
x − b
(y) i. m = y − a
x − b
(a)
j. m = y − a
x − b (x) k. C = 2π
r (r) l. f = ax + by (x)
m. s = ut + 1
2
at2
(a) n. F = GMm
r2 (G)
Understanding
The cost to rent a car is given by the formula C = 50d + 0.2k, where d = the number of days rented and
k = the number of kilometres driven. Lin has $300 to spend on car rental for her 4-day holiday. How far
can she travel on this holiday?
A cyclist pumps up a bike tyre that has a slow leak. The volume of air (in cm3
) after t minutes is given
by the formula:
V = 24000 − 300t
a. What is the volume of air in the tyre when it is first filled?
b. Write an equation and solve it to work out how long it takes the tyre to go completely flat.
Reasoning
- The total surface area of a cylinder is given by the formula T = 2πr2 + 2πrh, where r = radius and
h = height. A car manufacturer wants the engine’s cylinders to have a radius of 4 cm and a total
surface area of 400 cm2
. Show that the height of the cylinder is approximately 11.92 cm, correct to
2 decimal places. (Hint: Express h in terms of T and r.)
If B = 3x − 6xy, write x as the subject. Explain the process by showing all working.
Problem solving
Use algebra to show that 1
v = 1
u − 1
f
can also be written as u = fv
v + f
.
Consider the formula d = √b2 − 4ac.
Rearrange the formula to make a the subject.
Find values for a and b, such that:
4
x + 1 − 3
x + 2 = ax + b
(x + 1)(x + 2)
Reflection
How does rearranging formulas differ to solving linear equations?
CHALLENGE 4.2
The volume, V, of a sphere can be calculated using the formula V = 4
3
πr 3, where r is the radius of the sphere.
What is the radius of a spherical ball that has the capacity to hold 5 litres of water?
4.7 Review
4.7.1 Review questions
Fluency
The linear equation represented by the sentence ‘When a certain number is multiplied by 3, the result
is 5 times the certain number plus 7’ is:
a. 3x + 7 = 5x b. 5(x + 7) = 3x c. 5x + 7 = 3x d. 5x = 3x + 7
The solution to the equation x
3 = 5 is:
a. x = −15 b. x = 15 c. x = 1 2
3 d. x = 3
- What is the solution to the equation 7 = 21 + x?
a. x = 28 b. x = −28 c. x = −14 d. x = 14
- What is the solution to the equation 5x + 3 = 37?
a. x = 8 b. x = −8 c. x = 6.8 d. x = 106
- The solution to the equation 8 − 2x = 22 is:
a. x = 11 b. x = 15 c. x = −15 d. x = −7
- The solution to the equation 4x + 3 = 7x − 33 is:
a. x = −12 b. x = 12 c. x = 36
11
d. x = 30
11
- The solution to the equation 7(x − 15) = 28 is:
a. x = 11 b. x = 19 c. x = 20 d. x = 6.14
- When rearranging y = ax + b in terms of x, we obtain:
a. x = y − a
b
b. x = y − b
a
c. x = b − y
a
d. x = y + b
a
- Which of the following are linear equations?
a. 5x + y2 = 0 b. 2x + 3 = x − 2 c.
x
2 = 3
d. x2 = 1 e.
1
x
- 1 = 3x f. 8 = 5x − 2
g. 5(x + 2) = 0 h. x2 + y = −9 i. r = 7 − 5(4 − r)
- Solve each of the following linear equations.
a. 3a = 8.4 b. a + 2.3 = 1.7 c.
b
21 = −0.12
d. b − 1.45 = 1.65 e. b + 3.45 = 0 f. 7.53b = 5.64
- Solve each of the following linear equations.
a.
2x − 3
7 = 5 b. 5 − x
2 = −4 c. −3x − 4
5 = 3
d. 6
x = 5 e.
4
x = 3
5
f. x + 1.7
2.3 = −4.1
Solve each of the following linear equations.
a. 5(x − 2) = 6 b. 7(x + 3) = 40 c. 4(5 − x) = 15
d. 6(2x + 3) = 1 e. 4(x + 5) = 2x − 5 f. 3(x − 2) = 7(x + 4)
Liz has a packet of 45 Easter eggs. She saves 21 to eat tomorrow but rations the remainder so that she
can eat 8 eggs each hour.
a. Write a linear equation in terms of the number of hours, h, to represent this situation.
b. Work out how many hours it will take to eat today’s share.
Solve each of the following linear equations.
a. 11x = 15x − 2 b. 3x + 4 = 16 − x c. 5x + 2 = 3x + 8
d. 8x − 9 = 7x − 4 e. 2x + 5 = 8x − 7 f. 3 − 4x = 6 − x
Translate these sentences into algebraic equations.
Use x for the certain number.
a. Twice a certain number is equal to 3 minus that certain number.
b. When 8 is added to 3 times a certain number, the result is 19.
c. Multiplying a certain number by 6 equals 4.
d. Dividing 10 by a certain number is one more than dividing that number by 6.
e. Multiply a certain number by 2, then add 5. Multiply this result by 7. This expression equals 0.
f. Twice the distance travelled is 100 metres more than the distance travelled plus 50 metres.
Samuel decides to go on a holiday. He travels a certain distance on the first day, twice that distance on
the second day, three times that distance on the third day and four times that distance on the fourth
day. If his total journey is 2000 km, how far did he travel on the third day?
For each of the following, make the variable shown in brackets the subject of the formula.
a. y = 6x − 4 (x) b. y = mx + c (x) c. q = 2(P − 1) + 2r (P)
d. P = 2l + 2w (w) e. v = u + at (a) f. s = (
u + v
2 )t (t)
g. v2 = u2 + 2as (a) h. 2A = h(a + b) (b)
Problem solving
- John is comparing two car rental companies, Golden Ace Rental Company and Silver Diamond Rental
Company. Golden Ace Rental Company charges a flat rate of $38 per day and $0.20 per kilometre. The
Silver Diamond Rental Company charges a flat rate of $30 per day plus $0.32 per kilometre. John
plans to rent a car for three days.
a. Write an algebraic equation for the cost of renting a car for three days from the Golden Ace Rental
Company in terms of the number of kilometres travelled, k.
b. Write an algebraic equation for the cost of renting a car for three days from the Silver Diamond
Rental Company in terms of the number of kilometres travelled, k.
c. How many kilometres would John have to travel so that the cost of hiring from each company is the
same?
- Frederika has $24000 saved for a holiday and a new stereo. Her travel expenses are $5400 and her
daily expenses are $260.
a. Write down an equation for the cost of her holiday if she stays for d days.
Upon her return from holidays, Frederika wants to purchase a new stereo system that will
cost her $2500.
b. How many days can she spend on her holiday if she wishes to purchase a new stereo upon her
return?
- A maker of an orange drink purchases her raw materials from two sources. The first source provides
liquid with 6% orange juice, while the second source provides liquid with 3% orange juice. She
wishes to make 1litre of drink with 5% orange juice. Let x = amount of liquid (in litres) purchased
from the first source.
a. Write an expression for the amount of orange juice from the first supplier, given that x is the amount
of liquid.
b. Write an expression for the amount of liquid from the second supplier, given that x is the amount of
liquid used from the first supplier.
c. Write an expression for the amount of orange juice from the second supplier.
d. Write an equation for the total amount of orange juice in the mixture of the 2 supplies, given that
1 litre of drink is mixed to contain 5% orange juice.
e. How much of the first supplier’s liquid should she use?
- Rachel, the bushwalker, goes on a 4-day journey. She travels a certain distance on the first day, half
that distance on the second day, a third that distance on the third day and a fourth of that distance on
the fourth day. If the total journey is 50 km, how far did she walk on the first day?
- Svetlana, another bushwalker goes on a 5-day journey, using the same pattern as Rachel in the
previous question (a certain amount, then half that amount, then one third, one fourth and one fifth).
If her journey is also 50 km, how far did she travel on the first day?
- Nile.com, the Internet bookstore, advertises its shipping cost to Australia as a flat rate of $20 for up to
10 books; while Sheds & Meager, their competitor, offers a rate of $12 plus $1.60 per
if anyone os bored
