college algebra

Question

A waitress sold 17 ribeye steak dinners and 26 grilled salmon​ dinners, totaling ​$572.65 on a particular day. Another day she sold 26 ribeye steak dinners and 13 grilled salmon​ dinners, totaling ​$580.73. How much did each type of dinner​ cost?

Answer 1

If you bought 5 items at 3 dollars apiece, you would spend 5(3 dollars) =15 dollars. So is you bought 5 items at x dollars apiece, you would spend 5x dollars.

Let R = the cost of a ribeye steak dinner. Let G = the cost of a grilled salmon dinner.

On one day the waitress sold 17 ribeye steak dinners and 26 grilled salmon dinners for $572.65 That means that 17R + 26G = 572.65.

On another day, the waitress sold 26 ribeye steak dinners and 13 grilled salmon dinners for $580.73.

That means that 26R + 13G = 580.73.

Because these numbers are pretty complicated, the easiest way to solve them is to use what's called the addition method. So we write one equation over the other equation so the variables line up:

17R + 26G = 572.65

26R + 13G = 580.73

Now we want to change the equations so that if we add them up, one of the variables will disappear. To make R disappear, we need the least common multiple of 17 and 26. To make G disappear, we need to find the least common multiple of 26 and 13. That's easier since 26 X1 =26 and 13 X 2 = 26. So 26 is the least common mutiple of 26 and 13.What we want in the G column is to create 26G in the G column of one equation and -26G in the G column of the other equation. Then we can add the equations and the G's will disappear.

We already have 26G in the first equation, so let's leave that equation alone: 17R +26G = 572.65.

We have 13G in the second equation, but we want -26G. To get that we need to multiply 13G by -2. But to do that, we must multiply every term of the second equation by -2:

-2 (26R + 13G = 580.73) is -52R -26G = -1161.46.

Now we add up the equations:

17R + 26G = 572.65

-52R - 26G = -1161.46

-35R = -588.81

-35 -35

R = about $16.82

Now plug in the value of R into either one of the two equations in order to find G. To avoid rounding error, I'll 588.31/35 for R. I'll plug into the top equation:

17R + 26G = 572.65

17(16.82)+ 26G = 572.65

285.94 + 26G = 572.65

-285.94 -285.94

26G = 286.71

26 26

We get G = about $11.03

Check:

17R +26G = 572.65

17(16.82) + 26(11.03) = 572.72 Difference due to rounding error.

26R + 13G =580.63

26(16.82) + 13(11.03) = 580.71 Difference due to rounding error

To lessen the rounding error, learn how to store numbers in the memory of your calculator and then restore them as needed.

Answer 2

Let: x = steak

y = salmon

Then we have:

17x + 26y = 572.65

26x + 13y = 580.73

Now we have a system of equations we can solve using elimination or substitution. Let's do by elimination.

We multiply the second equation by -2 to get:

-52x + -26y = -1161.46

Then we set them up as such:

17x + 26y = 572.65

-52x + -26y = -1161.46

and add them both to get:

-35x = -588.81

Solving for x:

x = 16.82

Now we know x so we plug it into one of the equations to get y:

17(16.82) + 26y = 572.65

Solving for y:

y = 11.03

Plugging x and y back into the original equations will result in the approximate total since rounding contributes to the final answer.