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Middle Years

10.01 Linear rules

Worksheet
Linear equations
1

For each of the following statements:

i

Express as an equation.

ii

State whether the equation is linear.

a

y is equal to 7 less than 2 groups of x.

b

y equals x divided by 2 plus 8.

c

y equals - 1 divided by x plus 6.

d

y plus the quotient of x and the square of - 4 is equal to 14.

Tables of values
2

For each of the following tables representing a linear relationship:

i

State what happens to the y-value when the x-value increases by 1.

ii

Describe the rule between x and y in words.

iii

Write the linear equation for the rule between x and y.

a
x12345
y612182430
b
x12345
y89101112
c
x12345
y-7-8-9-10-11
d
x12345
y-2-6-10-14-18
e
x12345
y79111315
3

Consider the following table:

a

For every 1 unit increase in the x-value, is there a consistent change in the \\y-value?

b

State the change in y for every 1 unit increase in x.

x9182736
y-68-131-194-257
c

Explain why we can say that the relationship between x and y is linear.

d

Write the linear equation that describes the relationship between x and y.

e

Explain how the answer to part (b) helps in writing the rule for the linear relationship.

4

For each of the following tables, state whether the relationship between x and y is linear:

a
x12345
y48121620
b
x12345
y45678
c
x12478
y58111417
d
x12345
y-1371115
e
x12345
y51-3-7-11
f
x12345
y57131924
g
x127913
y-3-3-3-3-3
h
x510152025
y1530456075
i
x-1012
y71173
j
x-1012
y371115
k
x-1012
y3333
l
x-1012
y371113
m
x3333
y0369
n
x0369
y-70714
o
x0137
y1470-7
p
x-6-5-31
y091827
q
x-6-306
y271890
r
x-6-303
y091827
5

For each of the following tables:

i

Use the pattern to find the value of y, when x = 0.

ii

Write the linear equation that describes the relationship between x and y.

iii

Hence, complete the table.

a
x1234514
y48121620
b
x1234530
y-11357
c
x-1612345
y-2-4-6-8-10
d
x-3012345
y5244362820
e
x1234530
y-13-20-27-34-41
6

Given that the linear relationship between x and y is in the form y = m x + c, for each of the following tables:

i

Find the values of m and c.

ii

Write the linear equation that describes the relationship between x and y.

iii

Hence, complete the table.

a
x01234521
y0246810
b
x01234529
y81318232833
c
x01234521
y-21-16-11-6-14
d
x01234565
y24211815129
7

Find the linear equation between x and y for each of the following tables:

a
x-10123
y52-1-4-7
b
x12345
y710131619
c
x-8-7-6-5-4
y-36-31-26-21-16
d
x34567
y-12-17-22-27-32
Applications
8

The amount of medication M (in milligrams) in a patient’s body gradually decreases over time t (in hours) according to the equation M = 1050 - 15 t.

a

After 61 hours, how many milligrams of medication are left in the body?

b

How many hours will it take for the medication to be completely removed from the body?

9

A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth of the diver over 5 minutes:

\text{Number of minutes passed }(x)01234
\text{Depth of diver in metres }(y)01.42.84.25.6
a

What is the increase in depth each minute?

b

Write an equation for the relationship between the number of minutes passed (x) and the depth (y) of the diver.

c

In the equation from part (b), what does the number in front of the x represent?

d

Find the depth of the diver after 6 minutes.

e

How long does the diver takes to reach 12.6 metres beneath the surface?

10

After Mae starts running, her heartbeat, in beats per minutes, increases at a constant rate.

a

Write down the missing value from the table:

\text{Number of minutes passed }(x)024681012
\text{Heart rate }(y)495561677379
b

What is Mae's resting heart rate?

c

Find the change in y for every increase of one minute.

d

Form an equation that describes the relationship between the number of minutes passed, x, and Mae’s heartbeat, y.

e

In the equation from part (d), what does the number in front of the x represent?

f

Find Mae’s heartbeat after twenty minutes.

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