Writing the rule y = mx + c

Writing a linear rule in the form y = mx + c from a description, from a graph, or from two data points, using letters that suit the context.

Once you know the rate and the starting value, you can write the rule. The rule is more useful than the graph: it gives exact answers for any input, and you can type it into a spreadsheet.

The gradient–intercept form

Gradient–intercept form

$$y = mx + c \qquad m = \text{gradient (rate)}, \quad c = \text{vertical intercept (start)}$$

In context we swap $x$ and $y$ for letters that say what they are. A taxi fare is $F = 2.3d + 4.2$: $F$ dollars for $d$ kilometres, $\$2.30$ per km, $\$4.20$ to start.

From a description

Example 1●

A gym charges a $\$40$ joining fee and then $\$15$ a week. Write a rule for the total cost $C$ after $w$ weeks.

$m = 15$

The amount per week is the rate, so it multiplies w.

$c = 40$

The joining fee is paid at week 0: the starting value.

$C = 15w + 40$

Put them in the form, with the context letters.

From two data points

Often you don't see the intercept directly. You have two results, such as two bills. Find the gradient from the two points first, then substitute one point to find $c$.

Example 2●●

A phone repair shop charged $\$170$ for a $2$-hour job and $\$335$ for a $5$-hour job. Assuming a linear rule, write $C$ in terms of the hours $h$, and explain the intercept.

$m = \dfrac{335 - 170}{5 - 2} = \dfrac{165}{3} = 55$

Extra 165 dollars for 3 extra hours.

$170 = 55 \times 2 + c$, so $c = 170 - 110 = 60$

Substitute the point h = 2, C = 170 into C = 55h + c.

$C = 55h + 60$

Check with the other point: 55 × 5 + 60 = 335. It works.

The $\$60$ is a fixed fee charged for every job, such as a diagnostic fee.

The intercept is the cost for 0 hours.

InteractiveFind the rule that fits the receipts

Try this

Choose the repair shop. Adjust $m$ and $c$ until the line passes through both receipts. Your rule should match Example 2. Then try the electricity story and the bushwalk, where the gradient is negative. Write each rule using the context letters.

From a graph

Read $c$ where the line crosses the vertical axis. Pick two points that sit exactly on grid lines, find the rise over the run, and write the rule. Make sure you read the scales: one grid square might be $5$ minutes across and $\$20$ up.

Example 3●●

A graph shows the water left in a $9000$ L rainwater tank in a dry spell. It crosses the vertical axis at $9000$ and passes through $(30, 6000)$, with time in days. Write the rule and find when the tank is empty.

$m = \dfrac{6000 - 9000}{30 - 0} = -100$

Rise over run.

$W = -100t + 9000$, or $W = 9000 - 100t$

The intercept is 9000.

$0 = 9000 - 100t$, so $t = 90$ days

Empty means W = 0.

Your turn

An electricity bill has a fixed supply charge plus a cost per kWh. Using $400$ kWh cost $\$232$, and using $650$ kWh cost $\$322$. Write the rule for the bill $B$ in terms of usage $E$.

Show the working

$m = \dfrac{322 - 232}{650 - 400} = \dfrac{90}{250} = 0.36$, so $\$0.36$ per kWh. Then $232 = 0.36 \times 400 + c$, so $c = 232 - 144 = 88$. The rule is $B = 0.36E + 88$: a $\$88$ supply charge plus $36$ cents per kWh.

Flashcards

In $y = mx + c$

$m$ is the gradient (rate), $c$ is the vertical intercept (start)

Rule from two points

Find $m$ from the two points, then substitute one point to find $c$, then check with the other

Practice

What is the gradient of $y = 7 - 2x$?
What is the vertical intercept of $y = 7 - 2x$?
A tutor charges $\$20$ for materials and $\$55$ per lesson. Which rule gives the cost $C$ for $n$ lessons?
A line passes through $(2, 9)$ and $(6, 19)$. Find its gradient.
For the same line, find the vertical intercept $c$.
A repair shop charges by $C = 55h + 60$. What is the fixed fee, in dollars?
A tank's volume is $W = 9000 - 100t$ litres after $t$ days. After how many days is it empty?
Bills of $\$232$ for $400$ kWh and $\$322$ for $650$ kWh lie on a straight line. What is the cost per kWh, in dollars?
Exam style: A photographer charges $\$345$ for a $4$-hour shoot and $\$570$ for a $9$-hour shoot. Assuming a linear rule, find the hourly rate in dollars.
Exam style: For the same photographer, find the fixed fee in dollars.