6.01 Linear rules
Write an equation for y in terms of x for the following relationships:
The value of y is six less than the value of x.
The value of y is three times the value of x.
The value of y is five more than two times x.
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.
Yvonne has constructed the first three triangles of a pattern using matchsticks:
Yvonne makes a table comparing the number of triangles to the number of matchsticks needed to construct them as shown:
| Number of triangles | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Matchsticks | 3 | 6 |
Complete the table for the above pattern.
State the number of matchsticks that are used to make each new triangle.
Describe the relationship between the number of triangles and the number of matchsticks Yvonne will need.
Write the linear relationship for the number of matchsticks, M, in terms of the number of triangles, T, for this pattern.
Peter is making a sequence of shapes out of matchsticks:
Peter makes a table comparing the figure number to the number of matchsticks needed to construct it as shown:
| Figure number | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Matchsticks | 5 | 6 |
Complete the table for the above pattern.
State the number of extra matchsticks that are required to make the next figure.
Describe the relationship between the figure number and the number of matchsticks used to make it.
Write the algebraic rule for the number of matchsticks, M, in terms of the figure number, F, for this pattern.
Dave is constructing a continuing pattern of squares using matchsticks:
Dave made a table comparing the figure number to the number of matchsticks required.
| Figure no. | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Matchsticks | 4 | 7 | 10 | 13 |
If Dave wanted to continue the pattern, determine the number of matchsticks he would need for each square he adds.
Using the table Dave made, describe any patterns that you notice.
Describe the relationship between the figure number and the number of matchsticks it requires.
Write the algebraic rule for the number of matchsticks, M, in terms of the figure number, S, for this pattern.
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 |
Gwen sells bananas in bunches of 4.
Complete the following table:
Describe the relationship between the number of bunches and the number of bananas in words.
| \text{Bunches}\ (x) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Bananas}\ (y) |
Write an equation for the number of bananas, y, in terms of the number of bunches, x.
Tarek buys some decks of playing cards that contain 52 cards each.
Complete the following table:
Describe the relationship between the number of decks and the number of cards in words.
| \text{Decks} \ (d) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \text{Cards} \ (c) |
Write an equation for the number of cards, c, in terms of the number of decks, d.
Huda opens a bank account and deposits \$300. At the end of each week she adds \$10 to her account.
Complete the following table which shows the balance of Huda's account over the first four weeks:
| \text{Week }(W) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \text{Account total }(A) | \$300 | \$310 |
Write the linear relationship for Huda's account total, A, in terms of the number of weeks W, for which she has been adding to her account.
Hence find the amount of money in Huda's account after twelve weeks.
James already owns 5 marbles. He then buys some bags of marbles containing 4 marbles each.
Describe the relationship between the number of marbles James will have in total, and the number of bags of marbles he buys.
Write the algebraic rule for the number of marbles James will own, y, in terms of the number of bags he buys, x.
If James buys seven bags of marbles, how many marbles will he now own?
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.