13.09 Linear rules
For each of the following statements:
Express as an equation.
State whether the equation is linear.
y is equal to 7 less than 2 groups of x.
y equals x divided by 2 plus 8.
y equals - 1 divided by x plus 6.
y plus the quotient of x and the square of - 4 is equal to 14.
For each of the following tables representing a linear relationship:
State what happens to the y-value when the x-value increases by 1.
Describe the rule between x and y in words.
Write the linear equation for the rule between x and y.
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 6 | 12 | 18 | 24 | 30 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 8 | 9 | 10 | 11 | 12 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | -7 | -8 | -9 | -10 | -11 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | -2 | -6 | -10 | -14 | -18 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 7 | 9 | 11 | 13 | 15 |
Consider the following table:
For every 1 unit increase in the x-value, is there a consistent change in the \\y-value?
State the change in y for every 1 unit increase in x.
| x | 9 | 18 | 27 | 36 |
|---|---|---|---|---|
| y | -68 | -131 | -194 | -257 |
Explain why we can say that the relationship between x and y is linear.
Write the linear equation that describes the relationship between x and y.
Explain how the answer to part (b) helps in writing the rule for the linear relationship.
For each of the following tables, state whether the relationship between x and y is linear:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 4 | 8 | 12 | 16 | 20 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 4 | 5 | 6 | 7 | 8 |
| x | 1 | 2 | 4 | 7 | 8 |
|---|---|---|---|---|---|
| y | 5 | 8 | 11 | 14 | 17 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | -1 | 3 | 7 | 11 | 15 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 5 | 1 | -3 | -7 | -11 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 5 | 7 | 13 | 19 | 24 |
| x | 1 | 2 | 7 | 9 | 13 |
|---|---|---|---|---|---|
| y | -3 | -3 | -3 | -3 | -3 |
| x | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|
| y | 15 | 30 | 45 | 60 | 75 |
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 7 | 11 | 7 | 3 |
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 3 | 7 | 11 | 15 |
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 3 | 3 | 3 | 3 |
| x | -1 | 0 | 1 | 2 |
|---|---|---|---|---|
| y | 3 | 7 | 11 | 13 |
| x | 3 | 3 | 3 | 3 |
|---|---|---|---|---|
| y | 0 | 3 | 6 | 9 |
| x | 0 | 3 | 6 | 9 |
|---|---|---|---|---|
| y | -7 | 0 | 7 | 14 |
| x | 0 | 1 | 3 | 7 |
|---|---|---|---|---|
| y | 14 | 7 | 0 | -7 |
| x | -6 | -5 | -3 | 1 |
|---|---|---|---|---|
| y | 0 | 9 | 18 | 27 |
| x | -6 | -3 | 0 | 6 |
|---|---|---|---|---|
| y | 27 | 18 | 9 | 0 |
| x | -6 | -3 | 0 | 3 |
|---|---|---|---|---|
| y | 0 | 9 | 18 | 27 |
For each of the following tables:
Use the pattern to find the value of y, when x = 0.
Write the linear equation that describes the relationship between x and y.
Hence, complete the table.
| x | 1 | 2 | 3 | 4 | 5 | 14 |
|---|---|---|---|---|---|---|
| y | 4 | 8 | 12 | 16 | 20 |
| x | 1 | 2 | 3 | 4 | 5 | 30 |
|---|---|---|---|---|---|---|
| y | -1 | 1 | 3 | 5 | 7 |
| x | -16 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | -2 | -4 | -6 | -8 | -10 |
| x | -30 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| y | 52 | 44 | 36 | 28 | 20 |
| x | 1 | 2 | 3 | 4 | 5 | 30 |
|---|---|---|---|---|---|---|
| y | -13 | -20 | -27 | -34 | -41 |
Given that the linear relationship between x and y is in the form y = m x + c, for each of the following tables:
Find the values of m and c.
Write the linear equation that describes the relationship between x and y.
Hence, complete the table.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 21 |
|---|---|---|---|---|---|---|---|
| y | 0 | 2 | 4 | 6 | 8 | 10 |
| x | 0 | 1 | 2 | 3 | 4 | 5 | 29 |
|---|---|---|---|---|---|---|---|
| y | 8 | 13 | 18 | 23 | 28 | 33 |
| x | 0 | 1 | 2 | 3 | 4 | 5 | 21 |
|---|---|---|---|---|---|---|---|
| y | -21 | -16 | -11 | -6 | -1 | 4 |
| x | 0 | 1 | 2 | 3 | 4 | 5 | 65 |
|---|---|---|---|---|---|---|---|
| y | 24 | 21 | 18 | 15 | 12 | 9 |
Find the linear equation between x and y for each of the following tables:
| x | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | 5 | 2 | -1 | -4 | -7 |
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 7 | 10 | 13 | 16 | 19 |
| x | -8 | -7 | -6 | -5 | -4 |
|---|---|---|---|---|---|
| y | -36 | -31 | -26 | -21 | -16 |
| x | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|
| y | -12 | -17 | -22 | -27 | -32 |
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.
After 61 hours, how many milligrams of medication are left in the body?
How many hours will it take for the medication to be completely removed from the body?
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) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Depth of diver in metres }(y) | 0 | 1.4 | 2.8 | 4.2 | 5.6 |
What is the increase in depth each minute?
Write an equation for the relationship between the number of minutes passed (x) and the depth (y) of the diver.
In the equation from part (b), what does the number in front of the x represent?
Find the depth of the diver after 6 minutes.
How long does the diver takes to reach 12.6 metres beneath the surface?
After Mae starts running, her heartbeat, in beats per minutes, increases at a constant rate.
Write down the missing value from the table:
| \text{Number of minutes passed }(x) | 0 | 2 | 4 | 6 | 8 | 10 | 12 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate }(y) | 49 | 55 | 61 | 67 | 73 | 79 |
What is Mae's resting heart rate?
Find the change in y for every increase of one minute.
Form an equation that describes the relationship between the number of minutes passed, x, and Mae’s heartbeat, y.
In the equation from part (d), what does the number in front of the x represent?
Find Mae’s heartbeat after twenty minutes.