Linear Equations

Question

The manager of a restaurant found that the cost to produce 150 cups of coffee is ​$25.00, while the cost to produce 350 cups is ​$45.00. Assume the relationship between the cost y to produce x cups of coffee is linear.

A) Write a linear equation that expresses the cost, y, in terms of the number of cups of coffee, x.

B) How many cups of coffee are produced if the cost of production is $61.00?

Answer 1

A) Geometrically, you can think of the data given as two points: (150, 25) and (350, 45). The problem then reduces to finding the line between these points. Let this line be y = mx + b. Plug one of the points to get, 25 = 150m + b, and solve for b to get b = 25 - 150m. Plug the other point to get, 45 = 350m +b, use the value of b now, 45 = 350m + 25 - 150m = 25 + 200 m. You can solve for m now to get, m = (45 - 25) / 200 = 20 / 200 = 1/10.

Thus, we have found that our like looks like y = x / 10 + b. But, we know b = 25 - 150 m = 25 - 150 / 10 = 25 - 15 = 10. The line is thus, y = x / 10 + 10.

B) If the production cost is $61, then the number of cup of coffee is the value of x associated with y = 61. We use the equation of the line to get 61 = x / 10 + 10 => 61 - 10 = x / 10 => 51 * 10 = x => x = 510. So, the number of cups of coffee is 510.