Problem 1 :
A photocopy store charges $1.50 per copy for the first 10 copies and $1.00 per copy after the 10th copy. Let x be the number of copies, and let y be the total cost of photocopying (i) Draw graph of the cost as x goes from 0 to 50 copies (ii) Find the cost of making 40 copies.
Solution :
(i) :
(ii) :
x stands for number of copies and y stands for total cost.
y = 1.5x for 0 ≤ x ≤ 10
After 10th copy, cost per copy is $1.00.
Let us construct a function for the number of copies which is greater than 10.
y = 1.5(10) + 1(x - 10)
y = 15 + x - 10
y = 5 + x for x > 10
Substitute 40 for x.
y = x + 5
= 40 + 5
= 45
The cost of making 40 copies is $45.
Problem 2 :
Find at least two equations of the straight lines in the family of the lines y = 5x + b, for which b and the x-coordinate of the point of intersection of the lines with 3x - 4y = 6 are integers.
Solution :
Find the point of intersection of the above lines using substitution.
y = 5x + b
3x - 4y = 6
Substitute (5x + b) for y in 3x - 4y = 6.
3x - 4(5x + b) = 6
3x - 20x - 4b = 6
-17x - 4b = 6
-17x = 6 + 4b
x = -(6 + 4b)/17
Since x is an integer, (6 + 4b) has to be a multiple of 17.
6 + 4b = ±17, ±34,...............
6 + 4b = 17 ----> b = 11/4 (not an integer)
6 + 4b = -17 ----> b = -23/4 (not an integer)
6 + 4b = 34 ----> b = 7 (integer)
6 + 4b = -34 ----> b = -10 (integer)
The required equations are
y = 5x + 7
y = 5x - 10
Problem 3 :
Find all the equations of the straight lines in the family of the lines y = mx - 3, for which m and the x-coordinate of the point of intersection of the lines with x - y = 6 are integers.
Solution :
Find the point of intersection of the above lines using substitution.
y = mx − 3
x - y = 6
Substitute (mx - 3) for y in x - y = 6.
x - (mx - 3) = 6
x - mx + 3 = 6
x - mx = 3
x(1 - m) = 3
x = 3/(1 - m)
Find the possible integer values of m such that the value of x is also an integer.
m = -2 ----> x = 3/3 = 1
m = 0 ----> x = 3/1 = 3
m = 2 ----> x = 3/(-1) = -3
m = 4 ----> x = 3/(-3) = -1
The possible integer values of m are -2, 0, 2 and 4.
The required equations are
y = -2x - 3
y = -3
y = 2x - 3
y = 4x - 3
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