Lesson 8. Cover-up method for drawing straight lines

In this unit we have drawn equations of straight lines where y is on its own (examples: y = 2x + 1, y = 3x – 2)

If x and y are on the same side of the ‘=’ sign (examples: x + y = 2, 2x – 3y = 6), we can find the point to join up on a graph by using the cover-up method.

EXAMPLE:

Draw 2x + 3y = 6.

SOLUTION:

Use x = 0 so 2x = 0

Cover up 2x in the equation.

2x + 3y = 6 becomes 0 + 3y = 6 so y = 2

Use y = 0 so 3y = 0

Cover up 3y in the equation.

2x + 3y = 6 becomes 2x + 0 = 6 so x = 3

Always use x = 0 then y = 0.
Plot the points x = 0, y = 2 and x = 3, y = 0 on the graph and join them up to get your straight line.

EXAMPLE:

Suppose salami and sausage cost £6 and £3 per kilogram, and we wish to buy £12 worth. How much of each can we purchase?

SOLUTION:

Letting x and y be the weights of salami and sausage, the total cost is

6x + 3y = 12.

Solving for y gives the point-slope form

y = –2x + 4,

as above. That is, if we fist choose the amount of salami x, the amount of sausage can be computed as a function

y = f(x) = –2x + 4.

Since salami costs twice as much as sausage, adding one kilo of salami decreases the sausage by 2 kilos:

f(x + 1) = f(x) – 2, and the slope is –2.

The y-intercept point (x, y) = (0, 4) corresponds to buying only 4 kg of sausage; while the x-intercept point (x, y) = (2, 0) corresponds to buying only 2 kg of salami.

Note that the graph includes points with negative values of x or y, which have no meaning in terms of the original variables (unless we imagine selling meat to the butcher). Thus we should restrict our function f(x) to the domain 0 ≤ x ≤ 2.

Also, we could choose y as the independent variable, and compute x by the inverse linear function:

x = q(y) = – ¹/₂y + 2

over the domain 0 ≤ y ≤ 4.

Lesson 8

•  Test 1