Comparing equations question?

Question

Could someone point me in the direction of solving this, I am not necessarily looking for an answer, but I do not know where to start!

A vendor sells 216 regular coffees and 162 large coffees for a total of $783.

The question goes on to say that the next day 248 regular espressos and 186 large were sold. Is it possible that the vendor made $914.00 that day. How would you go about solving this?

Answer 1

you have a system of two equations:

you have 216R + 162L = 783

and 248R + 186L = 914

(R is the cost of a regular and L is the cost of a large -- I'm assuming that it's all coffee, not an espresso the second day...)

from the first part: R = (783 - 162L) / 216

substitute this into the second equationn:

248R + 186L = 914 becomes

248(783 - 162L) / 216 + 186L = 914

194184/216 - 40176L/216 + 186L = 914

899 - 186L + 186L = 914

899 = 914

this system has no solution, so it is not possible...

what that means is that the two lines:

216R + 162L = 783 and

248R + 186L = 914 are parallel, and never intersect

check: 248 / 216 = 31/27

186/162 = 31/27

914 / 783 can't be reduced

therefore, the ratio of the R to L terms (slope) is the same, but they have different "y" intercepts ==> parallel

check substituting the other way:

from the second equation:

L = (914 - 248R)/ 186

putting into the first equation

216R + 162(914 - 248R)/186 = 783

216R + 796.06 - 216R = 783

796.06 = 783 ===> no solution

By the way, 248 regulars and 186 larges would cost 31/27 * 783 = $899

Answer 2

Let X = price of regular

Let Y = price of large

216X + 162Y = 783

248X + 186Y = 914

See if the two equations/lines intersect.

Answer Below (if you want to know the solution):

(216X + 162Y = 783) x 186

(248X + 186Y = 914) x 162

40176X + 30132Y = 145638

40176X + 30132Y = 148068

*subtract 2 equations from each other

0 = -2430

Impossible. Thus, the vendor could not have made $914.

Answer 3

put it in a ratio form and solve it