
Answers to Questions on Linear Functions
 Find the slopes and the x and
yintercepts of the following lines.
 Question: y + 3 = 2 (x 
5)
Answer: Slope = 2; intercepts:
(7/2, 0) and (0, 7)
 Question: y = 1.2 x  7
Answer: Slope = 1.2, intercepts: (35/6, 0)
and (0, 7).
 Question: 3 x  5 y = 20
Answer: Slope = 3/5, intercepts: (20/3, 0)
and (0, 4).
 Question: y  c = 2 x + c/2
Answer: Slope = 2, intercepts: (3c/4, 0)
and (0, 3c/2).
Return to Exercises
 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).  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.  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!
Return to Exercises
 State whether the following pairs of lines
are parallel, perpendicular or neither:
 Question: y = (3/2)x 7
and 3x  2y = 4
Answer:
parallel; both have slopes 3/2
 Question: 5x  3y = 12 and 3x + 5y
= 10
Answer: perpendicular; slopes are
5/3 and 3/5.  Question: x  y =
10 and x + y = 1
Answer:
perpendicular; slopes are 1 and 1
 Question: x  2y = 1 and 2x  y = 5
Answer: neither; slopes are 1/2 and 2
 Question: x  3y = 5 and 2x + 6y
= 8
Answer: parallel; slopes are both
1/3  Question: 3x + 7y =
9 and 6x + 14y = 21
Answer:
neither, slopes are 3/7 and 3/7
 Question: y = (2/5)x + 2 and 5x  2y =
4
Answer: neither; slopes are 2/5
and 5/2  Question: x = 10
and y  10 = 0
Answer:
perpendicular; x = 10 is vertical, y = 10 is horizontal.
Return to Exercises
 Find the equation of each of the following
lines:
 Question: The
line with slope 1/2 and passing through the point (0,
3).
Answer: y = (1/2) x + 3
 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  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
 Question: The line with slope 6 and
passing through the graph of f(x) = x^{2} where x =
3.
Answer: y  9 = 6(x  3) or y = 6x
 9
 Question: The line passing through
(4,0) and the graph of f(x) = x^{2/3} where x =
8.
Answer: y  0 = (1/3)(x  4) or y
= (1/3)x + 4/3  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 Return to Exercises
 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.
 Question: Assume that y(t)
is linear. Using the data given, find the slope of
y(t).
Answer: Slope = 78.
 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.
 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. Return to Exercises
 Find the point(s) of intersection of each
of the following pairs of lines.
 Question: 2 x  y = 10 and x + y = 1
Answer: (3, 4)
 Question: y = 2 x + 5 and y  1 = 2 (x
3)
Answer: No solution; parallel
lines.
 Question: y = (2/3) x + 5 and 2 x 
3 y = 15
Answer: Infinitely many
solutions: {(c, (2/3)c + 5)  c is any real number}
 Question: 3 x + 3 y = 180 and 3.6 x
 3.6 y = 180
Answer: (55, 5)
Return to Exercises
