Linear Functions

Recognise a linear function $y = ax + b$ and state $a$ and $b$ , tell from a table of values whether a function is linear, find its rule, and use linear functions in real settings, forwards and backwards.

Before you start

A function machine applies the same rule to every input and gives exactly one output. The machine "multiply by $4$ , then subtract $1$ " turns the input $x$ into the output $y = 4x - 1$ : the input $5$ gives $5 \times 4 - 1 = 19$ .
A table of values has the values of $x$ in its first row and the corresponding values of $y$ in its second. Each column gives an ordered pair $(x, y)$ , $x$ first. For $y = x + 3$ , the values $x = 0$ , $1$ and $2$ give the pairs $(0, 3)$ , $(1, 4)$ and $(2, 5)$ , and these points lie on a straight line. Plotted points are joined only when the values in between make sense and the quantity changes steadily, not for a whole number of things such as people or trips.
To substitute a negative number, put it in brackets: if $x = -3$ , then $2x + 1 = 2(-3) + 1 = -6 + 1 = -5$ .
To solve an equation, do the same to both sides: from $2x + 1 = 11$ , subtract $1$ to get $2x = 10$ , then divide by $2$ to get $x = 5$ .

Linear Functions

The functions $y = x + 3$ , $y = 2x + 1$ and $y = 3x$ have the same form: the variable is multiplied by a number (in $y = x + 3$ the number is $1$ ), and a number is added (here $3$ , $1$ and $0$ ). Their points lie on straight lines. A function of this form is called a linear function.

Linear function

A linear function is a function that can be written in the form $y = ax + b$ , where $a$ and $b$ are numbers.

$a$ is the coefficient of $x$ (the number multiplying $x$ ) and $b$ is the constant term (the number on its own). Each may be positive, negative or zero.

When $x$ can take every value, the graph of a linear function is a straight line; when $x$ can only be a whole number (pens, trips, T-shirts), its points lie on a straight line and are not joined.

Examples: $y = 2x + 3$ has $a = 2$ and $b = 3$ . $y = -x + 5$ has $a = -1$ and $b = 5$ , since $-x$ means $-1 \times x$ . $y = 3x$ has $a = 3$ and $b = 0$ . $y = 4$ has $a = 0$ and $b = 4$ , since it is $y = 0x + 4$ .

Not linear: $y = x^{2}$ , because $x$ is squared, and $y = \frac{1}{x}$ , because a number is divided by $x$ . Their points do not lie on a straight line. The grid shows the points of $y = 2x + 1$ and of $y = x^{2}$ for $x = 0$ , $1$ , $2$ and $3$ .

A function machine whose steps are "multiply by a number", then "add or subtract a number", is a linear function: "multiply by $4$ , then subtract $1$ " is $y = 4x - 1$ , with $a = 4$ and $b = -1$ .

Example 1: Linear or Not?

Which of these are linear functions? For each linear one, state $a$ and $b$ .
(a) $y = 5x - 2$ (b) $y = x^{2}$ (c) $y = 7$ (d) $y = -3x$

Solution

(a) $y = 5x - 2$ is $y = 5x + (-2)$ . Linear, with $a = 5$ and $b = -2$ : the sign in front of a number belongs to it.

(b) $y = x^{2}$ has $x$ squared. Not linear.

(d) $y = -3x$ can be written as $y = -3x + 0$ . Linear, with $a = -3$ and $b = 0$ .

(a), (c) and (d) are linear, with $a = 5$ , $b = -2$ ; $a = 0$ , $b = 7$ ; and $a = -3$ , $b = 0$ . (b) is not linear.

Example 2: Writing a Function in the Form y = ax + b

Write each function in the form $y = ax + b$ , and state $a$ and $b$ .
(a) $y = 10 - 2x$ (b) $y = \frac{x}{3} + 4$ (c) $y = 3(x + 2)$

Solution

(a) The term in $x$ is $-2x$ , so $y = 10 - 2x$ can be written as $y = -2x + 10$ . So $a = -2$ and $b = 10$ (not $a = 10$ ).

(b) $\frac{x}{3}$ is $x$ divided by $3$ , which is $\frac{1}{3}$ of $x$ . So $y = \frac{x}{3} + 4$ can be written as $y = \frac{1}{3}x + 4$ , and $a = \frac{1}{3}$ , $b = 4$ . Here $x$ is divided by a number, which is allowed; in $y = \frac{1}{x}$ a number is divided by $x$ , which is not linear.

(c) Expand the brackets: $3(x + 2) = 3x + 6$ . So $y = 3(x + 2)$ can be written as $y = 3x + 6$ , and $a = 3$ , $b = 6$ .

(a) $y = -2x + 10$ : $a = -2$ , $b = 10$ . (b) $y = \frac{1}{3}x + 4$ : $a = \frac{1}{3}$ , $b = 4$ . (c) $y = 3x + 6$ : $a = 3$ , $b = 6$ .

Linear or Not from a Table, and Finding the Rule

For $y = 2x + 1$ , each time $x$ goes up by $1$ , $y$ goes up by $2$ ( $1$ , $3$ , $5$ , $7$ ): the steps in $y$ are equal, and the points lie on a straight line. For $y = x^{2}$ the steps in $y$ are $1$ , $3$ and $5$ ( $0$ , $1$ , $4$ , $9$ ): they are not equal, and the points do not lie on a straight line. This gives a test that needs no graph.

Linear or not, and the rule, from a table

When the values of $x$ in a table go up in equal steps, look at the steps in $y$ . If $y$ changes by the same amount at every step (up, down, or not at all), the pairs belong to a linear function and their points lie on a straight line. If not, the function is not linear.

To find the rule $y = ax + b$ when $x$ goes up by $1$ at each step: $a$ is the change in $y$ at each step (negative if $y$ goes down), and $b$ is the value of $y$ when $x = 0$ .

If $x$ goes up in equal steps of another size (such as $2$ or $10$ ), $a$ is the change in $y$ at each step divided by the change in $x$ (for steps of $2$ in $x$ and $10$ in $y$ , $a = 10 \div 2 = 5$ ).

Then check the rule with every column of the table.

Why $b$ is the value of $y$ when $x = 0$ : substituting $x = 0$ into $y = ax + b$ gives $y = a(0) + b = b$ . If the table has no column for $x = 0$ , substitute any column into $y = ax + b$ and solve for $b$ .

Example 3: Is the Table Linear?

Is each table linear? Give a reason.

(a)

$x$	0	1	2	3	4
$y$	3	5	7	9	11

(b)

$x$	1	2	3	4	5
$y$	1	4	9	16	25

(c)

$x$	1	2	3	4
$y$	10	7	4	1

Solution

(a) $x$ goes up by $1$ each time, and $y$ goes up by $2$ , $2$ , $2$ and $2$ . Equal steps: linear.

(b) $x$ goes up by $1$ each time, and $y$ goes up by $3$ , $5$ , $7$ and $9$ . The steps are not equal: not linear.

(a) and (c) are linear; (b) is not.

Example 4: Finding the Rule from a Table

Find the rule $y = ax + b$ for each table.

(a)

$x$	0	1	2	3	4
$y$	3	5	7	9	11

(b)

$x$	1	2	3	4
$y$	10	7	4	1

(c)

$x$	2	4	6	8
$y$	10	20	30	40

Solution

(a) $y$ goes up by $2$ for each step of $1$ in $x$ , so $a = 2$ . When $x = 0$ , $y = 3$ , so $b = 3$ . The rule is $y = 2x + 3$ .

Check every column: $2(1) + 3 = 5$ $\checkmark$ , $2(2) + 3 = 7$ $\checkmark$ , $2(3) + 3 = 9$ $\checkmark$ , $2(4) + 3 = 11$ $\checkmark$

(b) $y$ goes down by $3$ for each step of $1$ in $x$ , so $a = -3$ . There is no column for $x = 0$ , so substitute the first column, $x = 1$ and $y = 10$ , into $y = -3x + b$ :

$10 = -3(1) + b$

Add $3$ to both sides: $b = 13$ . The rule is $y = -3x + 13$ .

Check: $-3(2) + 13 = 7$ $\checkmark$ , $-3(3) + 13 = 4$ $\checkmark$ , $-3(4) + 13 = 1$ $\checkmark$

(c) Here $x$ goes up by $2$ at each step, and $y$ goes up by $10$ . For each $1$ that $x$ goes up, $y$ goes up by $10 \div 2 = 5$ , so $a = 5$ . Substitute the first column, $x = 2$ and $y = 10$ : $10 = 5(2) + b$ , so $b = 0$ . The rule is $y = 5x$ .

Check: $5(4) = 20$ $\checkmark$ , $5(6) = 30$ $\checkmark$ , $5(8) = 40$ $\checkmark$

(a) $y = 2x + 3$ (b) $y = -3x + 13$ (c) $y = 5x$

Linear Functions in Real Life

Many everyday charges are linear: a fixed charge, paid once, plus a charge for each unit used. The fixed charge is $b$ and the charge for each unit is $a$ .

A quantity that changes by the same amount in each minute or hour, such as the temperature of water heated steadily, is a linear function of the time too: $b$ is its value at the start and $a$ is its change in each unit of time.

Once the equation is written, it can be used forwards (substitute $x$ to find $y$ ) or backwards (substitute $y$ and solve the equation for $x$ ).

Example 5: Binding a Booklet

A printing shop in Bras Basah Complex charges $2 to bind a booklet and $0.10 for each page printed. The cost of a booklet of $n$ pages is $C$ dollars.
(a) Write a formula for $C$ in terms of $n$ , and say what $a$ and $b$ stand for.
(b) Find the cost of a booklet of $120$ pages.
(c) Priya paid $9.50 for her booklet. How many pages did it have?

Solution

(a) For $n$ pages the charge is $0.1 \times n = 0.1n$ dollars, and the binding charge of $2 is added once: $C = 0.1n + 2$ .

This is a linear function with $a = 0.1$ , the charge in dollars for each page, and $b = 2$ , the binding charge in dollars, paid once whatever the number of pages.

(b) Forwards: substitute $n = 120$ .

$C = 0.1(120) + 2 = 12 + 2 = 14$

$9.5 = 0.1n + 2$

Subtract $2$ from both sides:

$7.5 = 0.1n$

Divide both sides by $0.1$ :

$n = \frac{7.5}{0.1} = 75$

Check: $0.1(75) + 2 = 7.5 + 2 = 9.5$ $\checkmark$

(a) $C = 0.1n + 2$ . (b) A booklet of $120$ pages costs $14. (c) Priya's booklet had $75$ pages.

Example 6: Heating Water

In a Science lesson, Mei Ling heats a beaker of water and records its temperature every $2$ minutes.

Time, $t$ (minutes)	2	4	6	8
Temperature, $T$ (°C)	31	37	43	49

(a) Show that $T$ is a linear function of $t$ , and find the formula $T = at + b$ .
(b) Say what $a$ and $b$ stand for.
(c) If the water keeps heating at the same rate, after how many minutes will it reach $70$ °C?

Solution

(a) $t$ goes up by $2$ at each step, and $T$ goes up by $6$ each time: equal steps, so $T$ is a linear function of $t$ .

For each $1$ minute, $T$ goes up by $6 \div 2 = 3$ , so $a = 3$ .

There is no column for $t = 0$ , so substitute the first column, $t = 2$ and $T = 31$ , into $T = 3t + b$ :

$31 = 3(2) + b$

Subtract $6$ from both sides: $b = 25$ . The formula is $T = 3t + 25$ .

Check: $3(4) + 25 = 37$ $\checkmark$ , $3(6) + 25 = 43$ $\checkmark$ , $3(8) + 25 = 49$ $\checkmark$

(b) $a = 3$ : the temperature rises by $3$ °C every minute. $b = 25$ : the temperature at the start, when $t = 0$ , was $25$ °C.

$70 = 3t + 25$

Subtract $25$ from both sides:

$45 = 3t$

Divide both sides by $3$ : $t = 15$ .

Check: $3(15) + 25 = 45 + 25 = 70$ $\checkmark$

The formula holds only while the water heats at the same rate; it cannot be used beyond $100$ °C, where water boils.

(a) $T = 3t + 25$ . (b) The temperature rises by $3$ °C each minute, from $25$ °C at the start. (c) After $15$ minutes.

Common Mistakes

Taking $a$ to be the first number written. In $y = 10 - 2x$ , $a$ is $-2$ (the number multiplying $x$ , with its sign) and $b$ is $10$ .
Dropping the sign of $b$ . $y = 5x - 2$ is $y = 5x + (-2)$ , so $b = -2$ , not $2$ .
Forgetting the number that is not written. In $y = x + 3$ , $a = 1$ ; in $y = -x + 5$ , $a = -1$ ; in $y = 3x$ , $b = 0$ .
Calling $y = \frac{3}{x}$ linear because it looks like $y = \frac{x}{3}$ . Dividing $x$ by a number is linear; dividing a number by $x$ is not.
Taking the step in $y$ as $a$ when $x$ goes up by more than $1$ . If $x$ goes up by $2$ and $y$ by $10$ , then $a = 10 \div 2 = 5$ , not $10$ .
Checking a rule with only one or two columns of a table. The rule must fit every column.
In a setting, adding a fixed charge for every unit. A binding charge or a hire company's fixed charge is paid once: it is $b$ , not part of $a$ .

Practice Questions

Try each question on your own before you open its solution. The questions get harder as you go.

Question 1

State $a$ and $b$ for each linear function.
(a) $y = 4x + 7$ (b) $y = x - 6$ (c) $y = -2x + 1$
(d) $y = 5x$ (e) $y = -x - 3$

ShowHide solution

(a) $a = 4$ and $b = 7$ .

(b) $y = x - 6$ is $y = 1x + (-6)$ : $a = 1$ and $b = -6$ .

(d) $y = 5x$ is $y = 5x + 0$ : $a = 5$ and $b = 0$ .

(e) $-x$ is $-1 \times x$ , and $-3$ is added: $a = -1$ and $b = -3$ .

(a) $a = 4$ , $b = 7$ (b) $a = 1$ , $b = -6$ (c) $a = -2$ , $b = 1$ (d) $a = 5$ , $b = 0$ (e) $a = -1$ , $b = -3$

Question 2

Which of these are linear functions? For each linear one, write it in the form $y = ax + b$ and state $a$ and $b$ .
(a) $y = 6 + 4x$ (b) $y = \frac{x}{2} - 3$ (c) $y = 2(x - 5)$
(d) $y = -5$ (e) $y = \frac{3}{x}$ (f) $y = x^{2} + 1$

ShowHide solution

(a) $y = 4x + 6$ : linear, with $a = 4$ and $b = 6$ .

(b) $y = \frac{1}{2}x - 3$ : linear, with $a = \frac{1}{2}$ and $b = -3$ .

(d) $y = 0x - 5$ : linear, with $a = 0$ and $b = -5$ .

(e) $3$ is divided by $x$ : not linear.

(f) $x$ is squared: not linear.

(a), (b), (c) and (d) are linear; (e) and (f) are not.

Question 3

Write the rule of each function machine as an equation in the form $y = ax + b$ , and state $a$ and $b$ .

(a)

(b)

(d) Find the output of machine (a) and of machine (b) when the input is $3$ . Does the order of the steps matter?

ShowHide solution

(a) Multiply $x$ by $2$ : $2x$ . Then add $5$ : $y = 2x + 5$ . So $a = 2$ and $b = 5$ .

(b) Add $5$ to $x$ : $x + 5$ . Then multiply by $2$ : $y = 2(x + 5)$ . Expand the brackets: $y = 2x + 10$ . So $a = 2$ and $b = 10$ .

(c) Subtract $x$ from $8$ : $8 - x$ . Then divide by $2$ , which is the same as multiplying by $\frac{1}{2}$ : $y = \frac{1}{2}(8 - x)$ . Expand the bracket: $y = \frac{1}{2} \times 8 - \frac{1}{2} \times x = 4 - \frac{1}{2}x$ , which can be written as $y = -\frac{1}{2}x + 4$ . So $a = -\frac{1}{2}$ and $b = 4$ .

Check (c) with the input $2$ : $8 - 2 = 6$ , then $6 \div 2 = 3$ ; and $-\frac{1}{2}(2) + 4 = -1 + 4 = 3$ $\checkmark$

(d) Machine (a): $2(3) + 5 = 6 + 5 = 11$ . Machine (b): $3 + 5 = 8$ , then $8 \times 2 = 16$ . The outputs are different, so the order of the steps matters.

(a) $y = 2x + 5$ : $a = 2$ , $b = 5$ . (b) $y = 2x + 10$ : $a = 2$ , $b = 10$ . (c) $y = -\frac{1}{2}x + 4$ : $a = -\frac{1}{2}$ , $b = 4$ . (d) $11$ and $16$ : the order matters.

Question 4

(a) Copy and complete the table of values for $y = -4x + 1$ .

$x$	−2	−1	0	1	2
$y$	?	?	?	?	?

(b) By how much does $y$ change each time $x$ goes up by $1$ ? Compare it with $a$ .
(c) What is the value of $y$ when $x = 0$ ? Compare it with $b$ .

ShowHide solution

(a) $x = -2$ : $y = -4(-2) + 1 = 8 + 1 = 9$

$x = -1$ : $y = -4(-1) + 1 = 4 + 1 = 5$

$x = 0$ : $y = -4(0) + 1 = 0 + 1 = 1$

$x = 1$ : $y = -4(1) + 1 = -3$ ; $x = 2$ : $y = -4(2) + 1 = -7$

$x$	−2	−1	0	1	2
$y$	9	5	1	−3	−7

(b) $y$ goes down by $4$ each time: a change of $-4$ , which is $a$ .

(a) $9$ , $5$ , $1$ , $-3$ , $-7$ . (b) A change of $-4$ , the same as $a$ . (c) $1$ , the same as $b$ .

Question 5

For each table, say whether it is linear. If it is, find its rule $y = ax + b$ .

(a)

$x$	0	1	2	3	4
$y$	−4	−1	2	5	8

(b)

$x$	1	2	3	4
$y$	2	5	10	17

ShowHide solution

(a) $x$ goes up by $1$ each time and $y$ goes up by $3$ each time: linear, with $a = 3$ . When $x = 0$ , $y = -4$ , so $b = -4$ . The rule is $y = 3x - 4$ .

Check every column: $3(1) - 4 = -1$ $\checkmark$ , $3(2) - 4 = 2$ $\checkmark$ , $3(3) - 4 = 5$ $\checkmark$ , $3(4) - 4 = 8$ $\checkmark$

(b) $x$ goes up by $1$ each time, but $y$ goes up by $3$ , $5$ and $7$ : the steps are not equal, so it is not linear.

(a) Linear: $y = 3x - 4$ . (b) Not linear.

Question 6

Find the rule $y = ax + b$ for each table.

(a)

$x$	−2	−1	0	1	2
$y$	9	7	5	3	1

(b)

$x$	1	3	5	7
$y$	4	10	16	22

ShowHide solution

(a) $x$ goes up by $1$ and $y$ goes down by $2$ each time, so $a = -2$ . When $x = 0$ , $y = 5$ , so $b = 5$ . The rule is $y = -2x + 5$ .

Check: $-2(-2) + 5 = 4 + 5 = 9$ $\checkmark$ , $-2(-1) + 5 = 2 + 5 = 7$ $\checkmark$ , $-2(1) + 5 = 3$ $\checkmark$ , $-2(2) + 5 = 1$ $\checkmark$

(b) Here $x$ goes up by $2$ each time and $y$ goes up by $6$ . For each $1$ that $x$ goes up, $y$ goes up by $6 \div 2 = 3$ , so $a = 3$ .

There is no column for $x = 0$ , so substitute the first column, $x = 1$ and $y = 4$ , into $y = 3x + b$ :

$4 = 3(1) + b$

Subtract $3$ from both sides: $b = 1$ . The rule is $y = 3x + 1$ .

Check: $3(3) + 1 = 10$ $\checkmark$ , $3(5) + 1 = 16$ $\checkmark$ , $3(7) + 1 = 22$ $\checkmark$

(a) $y = -2x + 5$ (b) $y = 3x + 1$

Question 7

The table belongs to a linear function, but three of its values of $y$ are missing.

$x$	0	1	2	3	4
$y$	?	7	?	15	?

(a) Find the rule $y = ax + b$ .
(b) Complete the table.

ShowHide solution

(a) From $x = 1$ to $x = 3$ , $x$ goes up by $2$ and $y$ goes up from $7$ to $15$ , by $8$ . For each $1$ that $x$ goes up, $y$ goes up by $8 \div 2 = 4$ , so $a = 4$ .

Substitute $x = 1$ and $y = 7$ into $y = 4x + b$ : $7 = 4(1) + b$ , so $b = 3$ . The rule is $y = 4x + 3$ .

Check with the other given column: $4(3) + 3 = 12 + 3 = 15$ $\checkmark$

(b) $x = 0$ : $y = 4(0) + 3 = 3$ . $x = 2$ : $y = 4(2) + 3 = 11$ . $x = 4$ : $y = 4(4) + 3 = 19$ .

$x$	0	1	2	3	4
$y$	3	7	11	15	19

The steps in $y$ are all $4$ : $3$ , $7$ , $11$ , $15$ , $19$ $\checkmark$

(a) $y = 4x + 3$ . (b) The missing values are $3$ , $11$ and $19$ .

Question 8 (Setting: Bicycle Hire at East Coast Park)

At a bicycle kiosk in East Coast Park, hiring a bicycle costs a fixed charge of $4 plus $6 for each hour. The cost of hiring a bicycle for $h$ hours is $C$ dollars.
(a) Write a formula for $C$ in terms of $h$ .
(b) Find the cost of hiring a bicycle for $3$ hours.
(c) Wei Ming paid $34. For how many hours did he hire his bicycle?

ShowHide solution

(a) $h$ hours at $6 an hour cost $6h$ dollars, and the fixed charge of $4 is added once: $C = 6h + 4$ .

(b) Substitute $h = 3$ :

$C = 6(3) + 4 = 18 + 4 = 22$

$34 = 6h + 4$

Subtract $4$ from both sides:

$30 = 6h$

Divide both sides by $6$ : $h = 5$ .

Check: $6(5) + 4 = 30 + 4 = 34$ $\checkmark$

(a) $C = 6h + 4$ . (b) $22. (c) Wei Ming hired the bicycle for $5$ hours.

Question 9 (Setting: A New Freezer)

A new freezer in an air-conditioned shop is switched on. For the first $8$ hours, the temperature inside it, $T$ °C, $h$ hours after it is switched on is given by $T = 24 - 4h$ .
(a) State $a$ and $b$ for this linear function, and say what each means.
(b) Make a table of values for $h = 0, 1, 2, 3, 4, 5, 6, 7$ and $8$ . Plot the pairs $(h, T)$ , and say whether the points should be joined.
(c) What is the temperature after $3$ hours?
(d) After how many hours is the temperature $-8$ °C?

ShowHide solution

(a) $T = 24 - 4h$ can be written as $T = -4h + 24$ , so $a = -4$ and $b = 24$ .

$b = 24$ : the temperature is $24$ °C when the freezer is switched on ( $h = 0$ ). $a = -4$ : the temperature falls by $4$ °C every hour.

(b) $h = 0$ : $T = 24 - 4(0) = 24$ . $h = 1$ : $T = 24 - 4(1) = 20$ . Each hour after that the temperature is $4$ °C lower.

Time, $h$ (hours)	0	1	2	3	4	5	6	7	8
Temperature, $T$ (°C)	24	20	16	12	8	4	0	−4	−8

One square across stands for $1$ hour and one square up for $4$ °C.

The points lie on a straight line going down. Time and temperature take every value in between the readings, and the temperature falls steadily, so the points are joined.

(d) Substitute $T = -8$ and solve:

$-8 = 24 - 4h$

Add $4h$ to both sides:

$4h - 8 = 24$

Add $8$ to both sides:

$4h = 32$

Divide both sides by $4$ : $h = 8$ .

Check: $24 - 4(8) = 24 - 32 = -8$ $\checkmark$

(a) $a = -4$ (the temperature falls $4$ °C each hour) and $b = 24$ (it starts at $24$ °C). (b) The points lie on a straight line and are joined. (c) $12$ °C. (d) After $8$ hours.

Question 10 (Setting: Class T-shirts)

Class 1A is ordering printed T-shirts for Sports Day. The printing shop sends this table of prices.

Number of T-shirts, $n$	10	20	30	40
Cost, $C$ ($)	130	230	330	430

(a) Show that $C$ is a linear function of $n$ , and find the formula $C = an + b$ .
(b) What do $a$ and $b$ stand for?
(c) Find the cost of $35$ T-shirts.
(d) The class has $500 to spend. How many T-shirts can it order for exactly that amount?

ShowHide solution

(a) $n$ goes up by $10$ at each step and $C$ goes up by $100$ each time: equal steps, so $C$ is a linear function of $n$ .

For each extra T-shirt, $C$ goes up by $100 \div 10 = 10$ , so $a = 10$ .

Substitute the first column, $n = 10$ and $C = 130$ , into $C = 10n + b$ :

$130 = 10(10) + b$

Subtract $100$ from both sides: $b = 30$ . The formula is $C = 10n + 30$ .

Check: $10(20) + 30 = 230$ $\checkmark$ , $10(30) + 30 = 330$ $\checkmark$ , $10(40) + 30 = 430$ $\checkmark$

(b) $a = 10$ : each T-shirt costs $10. $b = 30$ : a fixed charge of $30, paid once for the order, however many T-shirts there are.

(d) Substitute $C = 500$ and solve:

$500 = 10n + 30$

Subtract $30$ from both sides:

$470 = 10n$

Divide both sides by $10$ : $n = 47$ .

Check: $10(47) + 30 = 470 + 30 = 500$ $\checkmark$

(a) $C = 10n + 30$ . (b) $10 for each T-shirt, plus a fixed charge of $30. (c) $380. (d) $47$ T-shirts.

Question 11 (Setting: An EZ-Link Card)

Mei Ling has $30 on her EZ-Link card, and each of her bus trips costs $1.50. After $n$ trips, the amount left on the card is $B$ dollars.
(a) Write a formula for $B$ in terms of $n$ , and state $a$ and $b$ .
(b) How much is left after $8$ trips?
(c) After how many trips is $12 left?
(d) Can the amount left ever be exactly $20.50? Explain.
(e) If the pairs $(n, B)$ were plotted, should the points be joined?

ShowHide solution

(a) Each trip takes $1.50 off the $30, so after $n$ trips $1.5n$ dollars have been taken off: $B = 30 - 1.5n$ , which can be written as $B = -1.5n + 30$ . So $a = -1.5$ and $b = 30$ .

(b) Substitute $n = 8$ :

$B = 30 - 1.5(8) = 30 - 12 = 18$

$12 = 30 - 1.5n$

Add $1.5n$ to both sides:

$1.5n + 12 = 30$

Subtract $12$ from both sides:

$1.5n = 18$

Divide both sides by $1.5$ : $n = 12$ .

Check: $30 - 1.5(12) = 30 - 18 = 12$ $\checkmark$

(d) Substitute $B = 20.5$ : $20.5 = 30 - 1.5n$ gives $1.5n = 9.5$ , so $n = \frac{9.5}{1.5} = 6\frac{1}{3}$ .

A number of trips must be a whole number, so the amount left can never be exactly $20.50. After $6$ trips $21 is left, and after $7$ trips $19.50.

(e) No. A number of trips is a whole number, so there is nothing to draw between the points.

(a) $B = 30 - 1.5n$ , with $a = -1.5$ and $b = 30$ . (b) $18. (c) After $12$ trips. (d) No: it would need $6\frac{1}{3}$ trips. (e) No.

Question 12 (Setting: Choosing a Phone Plan)

Priya is choosing between two mobile phone plans. Plan A costs $15 a month plus $2 for each GB of data used. Plan B costs $9 a month plus $3 for each GB of data used. In a month in which she uses $n$ GB, Plan A costs $A$ dollars and Plan B costs $B$ dollars.
(a) Write a formula for $A$ and a formula for $B$ in terms of $n$ .
(b) Which plan is cheaper in a month in which she uses $4$ GB?
(c) For how many GB do the two plans cost the same?
(d) Priya usually uses $10$ GB a month. Which plan should she choose, and how much does she save each month?

ShowHide solution

(a) The fixed monthly charge is added once and the charge for data is paid for each GB: $A = 2n + 15$ and $B = 3n + 9$ .

(b) Substitute $n = 4$ : $A = 2(4) + 15 = 8 + 15 = 23$ and $B = 3(4) + 9 = 12 + 9 = 21$ . Plan B is cheaper, by $2.

$2n + 15 = 3n + 9$

Subtract $2n$ from both sides:

$15 = n + 9$

Subtract $9$ from both sides: $n = 6$ .

Check: $A = 2(6) + 15 = 27$ and $B = 3(6) + 9 = 27$ $\checkmark$ Both plans cost $27 for $6$ GB.

(d) Substitute $n = 10$ : $A = 2(10) + 15 = 35$ and $B = 3(10) + 9 = 39$ . Plan A is cheaper: she saves $39 − $35 = $4 each month.

(a) $A = 2n + 15$ and $B = 3n + 9$ . (b) Plan B ($21, against $23 for Plan A). (c) $6$ GB. (d) Plan A, saving $4 a month.

Key Points

A linear function can be written in the form $y = ax + b$ , where $a$ and $b$ are numbers: $a$ is the coefficient of $x$ and $b$ is the constant term; either may be negative or zero.
To state $a$ and $b$ , first write the function in the form $y = ax + b$ : put the term in $x$ first, expand any brackets, and write $\frac{x}{3}$ as $\frac{1}{3}x$ .
A function is not linear when $x$ is squared or a number is divided by $x$ .
When $x$ can take every value, the graph of a linear function is a straight line; when $x$ can only be a whole number (pens, trips, T-shirts), its points lie on a straight line and are not joined.
When $x$ goes up in equal steps, a table is linear if $y$ changes by the same amount at every step.
$a$ is the change in $y$ divided by the change in $x$ ; $b$ is the value of $y$ when $x = 0$ , or is found by substituting a column and solving. Check the rule with every column.
In a setting, $b$ is the fixed amount, paid once or present at the start, and $a$ is the amount for each unit; substitute to go forwards, and solve an equation to go backwards.