Systems of Equations
The equations used in this problem to measure our income are y=6x+30 and y=.5x+250. In the terms of variables the y is total income and x is batches sold. I put this equation together by multiplying the total cost of a batch with the number of batches sold, 6x, and adding the base cost of $30 per day from the movie theater ad. The total cost is y= .5x+250. Y represents our total, found by multiplying .5 by x, (number of cupcakes sold) Add $250 for the daily cost of electricity etc.
First you solve for x and y. The first equation, y = 6x + 30, the second being, x = -500 + 2y (y = .5x + 250; subtract .5x from the right side of the equation [ -.5x + y = 250]; subtract y from the left side [-.5x = 250 - y]; divide by -.5 on both sides of the equation). Substitute the x in the equation, y = 6x + 30. This would be, y = 6(-500 + 2y) + 30. Then, solve the equation. The final equation should be y = -2970 +12y --> y = 270. We have our y value, now substitute the value into both equations in order to find the x value (270 = 6x + 30 240 = 6x 40 = x) After that the ultimate coordinate point that you should get is (40, 270).
Firstly take one of the coefficients of a variable opposite of that in the other equation. No two values canceled the other out, so I multiplied the equation y = .5 + 250 by -12. The equation becomes -12y = -6x - 3000. Both -6x and 6x cancel each other out making -11y = 2970. Divide both sides of the equation by -11 to single out y, making it y = 270. Finally, substitute the y (270) into both equations and solve. The answer should be the point (40, 270).
The equation was already in slope-intercept form so graphing was easy. Our equation representing profits would have a y-intercept of 30 and slope of 6. Our cost equation would have a y-intercept of 250 and a slope of .5. The point they touch is, (40,270) therefor its the solution.
Convert each equation into standard form initially. The equation y = 6x + 30 you would enter into the Matrix would include, -6, 1, and 30 and do the same for the other. Press the buttons, 2nd quit, 2nd Matrix, Math, 'rref', and finally 2nd Matrix. When you have reached the 2nd Matrix, press enter twice, and it shows [1 0 40] and [0 1 270]. This means an x value of 40 and a y value of 270, making the point of (40,270).
270$ for 40 cupcakes is the amount that we can sell per day and break evenly. For our company's well being, we need to make profit from our cupcakes. Therefore, in order to gain a profit, we need to sell at least forty-one batches of cupcakes. If we sell thirty-nine batches or less, we lose money.
What About Our Thumbs?
Our little fingers are safe this time. We can repay Gino with interest. We had a 180 day span, and we made of 96 batches of cupcakes a day, we start by multiplying 180 by 96 in order to find the total amount of batches of cupcakes made. Substitute this into both of the equations for x. The two equations would be, y = 6(17280) + 30 (income), and y = .5(17280) + 250 (costs). Our total would be $103,710, and our total costs would be $8,890. Once we subtract both of these in order to find our profit, we have made a total of $94,820. With Gino's loan of $50,000, and our interest of $8,500, we will be able to reimburse all of his money, while also gaining additional money of our own $36,320.