marj29kuletz Posts: 1, Reputation: 1 New Member #1 Aug 25, 2010, 04:45 AM
Problem solving involving linear function
The cost of food for a school party is partly constant and partly depends on the number of students. For 95 students ,the cost 5,652.50 ,and for 44 students ,it is 3,272.50. Find the cost for 48 students .express the cost of food as a function of the number of students.
 ebaines Posts: 12,130, Reputation: 1307 Expert #2 Aug 25, 2010, 06:18 AM

For linear equations the function is: y = mx+b, where y is the cost of the food, x is the number of students, m is the slope of the line and b is the y-axis intercept. You have been given two data points that sit on that line:

$
(x_1,y_1) = (44, 3272.5)\\
(x_2, y_2) = (95, 5652.5)
$

If you are given two data points that sit on the line, you can determine the function as follows:

1. First determine the value for m, the slope. Remember that slope is defined as "rise over run."

$
m = \frac {\Delta y} {\Delta x} = \frac {y_2-y_1} {x_2 - x_1}
$

2. Determine b. Since the equation is y=mx + b, use the values of (y_1,x_1) in that equation and solve for b:

$
y_1 = mx_1 +b \\
b = y_1 - mx_1
$

I'll give you an example so you can see how this works. Suppose:
$
(x_1,y_1) = (6, 5)\\
(x_2, y_2) = (4, 9)
$

Then:
$
m = \frac {\Delta y} {\Delta x} = \frac {y_2-y_1} {x_2 - x_1} = \frac {9-5}{4-6} = - 2 \\
b = y_1 - mx_1 = 5-(- 2 )6 = 5+12 = 17
$

hence:
$
y = -2x + 17
$

Post back with what you get for an answer.

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