# Answers to Questions on Linear Functions

1. Find the slopes and the x- and y-intercepts of the following lines.

1. Question: y + 3 = -2 (x - 5)

Answer: Slope = -2; intercepts: (7/2, 0) and (0, 7)

2. Question: y = 1.2 x - 7

Answer: Slope = 1.2, intercepts: (35/6, 0) and (0, -7).

3. Question: 3 x - 5 y = 20

Answer: Slope = 3/5, intercepts: (20/3, 0) and (0, -4).

4. Question: y - c = 2 x + c/2

Answer: Slope = 2, intercepts: (-3c/4, 0) and (0, 3c/2).

2. In economics the demand function relates the price per unit of an item to the number of units that consumers will buy at that price. The demand, q, is considered to be the independent variable, while the price, p, is considered to be the dependent variable.

Suppose that in a certain market, the demand function for widgets is a linear function

p = -0.75q + 54,

where p is the price in dollars and q is the number of units (hundreds widgets in this case).

1. Question: What is the slope of this function? Explain the meaning of the sign of the slope in practical terms.

Answer: The slope is -0.75. Since the slope is negative, the price decreases as the number of items bought by consumers increases. You could also say that the cheaper the item, the greater the demand.

2. Question: Find the p- and q- intercepts for this function. What is the significance of these intercepts in the context of the problem?

Answer: (0, 54); Since q = 0, consumers will buy no widgets when the price is \$54.
(72, 0); Since the price is zero, we can see that 7200 widgets could be given away for free!

3. State whether the following pairs of lines are parallel, perpendicular or neither:
1. Question: y = (3/2)x -7 and 3x - 2y = 4

Answer: parallel; both have slopes 3/2

2. Question: 5x - 3y = 12 and 3x + 5y = 10

Answer: perpendicular; slopes are 5/3 and -3/5.

3. Question: x - y = 10 and x + y = -1

Answer: perpendicular; slopes are 1 and -1

4. Question: x - 2y = 1 and 2x - y = 5

Answer: neither; slopes are 1/2 and 2

5. Question: x - 3y = 5 and -2x + 6y = 8

Answer: parallel; slopes are both 1/3

6. Question: 3x + 7y = 9 and -6x + 14y = 21

Answer: neither, slopes are -3/7 and 3/7

7. Question: y = (2/5)x + 2 and 5x - 2y = -4

Answer: neither; slopes are 2/5 and 5/2

8. Question: x = 10 and y - 10 = 0

Answer: perpendicular; x = 10 is vertical, y = 10 is horizontal.

4. Find the equation of each of the following lines:
1. Question: The line with slope -1/2 and passing through the point (0, 3).

Answer: y = (-1/2) x + 3

2. Question: The line with slope -2/3 and containing the point (6, -1).

Answer: y + 1 = (-2/3)(x - 6) or y = (-2/3)x + 3

3. Question: The line passing through the points (7, -1) and (4, 5).

Answer: y + 1 = -2(x - 7) or y - 5 = -2(x - 4) or y = -2x + 13

4. Question: The line with slope 6 and passing through the graph of f(x) = x2 where x = 3.

Answer: y - 9 = 6(x - 3) or y = 6x - 9

5. Question: The line passing through (4,0) and the graph of f(x) = x2/3 where x = -8.

Answer: y - 0 = (-1/3)(x - 4) or y = (-1/3)x + 4/3

6. Question: The line perpendicular to 3x + y = 17 and passing through (15, 2.5).

Answer: y - 2.5 = (1/3) (x - 15) or y = (1/3)x - 2.5

5. A small college has 2546 students in 1994 and 2702 students in 1996. Assume that the enrollment follows a linear growth pattern. Let t = 0 correspond to 1990 and let y(t) represent the enrollment in year t.

1. Question: Assume that y(t) is linear. Using the data given, find the slope of y(t).

2. Question: What does the slope of y(t) signify in terms of enrollment growth?

Answer: This means that the enrollment of the college is increasing by about 78 students per year.

3. Question: Find an equation for y(t) and use it predict the enrollment of the college in 1999.

Answer: y = 78t + 2234. In 1999 there will be about 2936 students at the college, provided this linear trend continues.

6. Find the point(s) of intersection of each of the following pairs of lines.

1. Question: 2 x - y = 10 and x + y = -1

2. Question: y = 2 x + 5 and y - 1 = 2 (x -3)