please help with math homework

Question

The waitress or 20 ribeye steak dinners and 18 grilled salmon dinners totaling $587.87 on particular day. another day she sold 30 ribeye steak dinners and six grilled salmon dinners totaling $580.34. How much did each type of dinner cost?

Answer 1

This problem can be solved by creating a system of equations. The two unknown elements in this problem are the cost of the ribeye dinner and the cost of the salmon dinner. Therefore, we need to define our unknowns as follows:

X= cost of ribeye steak dinner

Y=cost of grilled salmon dinners

20x+18y=587.87

30x+6y=580.34

There are several methods that can be used to solve and find the values of both x and y (substitution, elimination by addition, elimination by subtracting, graphing, etc).

I am going to demonstrate and solve this system of equation by using the elimination by subtraction method. The variable I want to eliminate is variable y.

20x+18y=587.87

30x+6y=580.34

To eliminate y by subtraction, I need the coefficient (basically the number in front of the variable) of y in both equations to be the same. Again, there is more than one approach to this. Since 18 is the coefficient of y in my first equation, I need to make 18 the coefficient of y in my second equation. I can do that by using the distributive property to multiply by second equation by 6. Please see below.

20x+18y=587.87-----> 20x+18y=587.87 ------> 20x+18y=587.87

30x+6y=580.34-----> 3(30x+6y)=3(580.34)------>90x+18y=1,741.02

Now, that both coefficients of y are 18, we can simply subtract to eliminate y.

20x+18y=587.87

90x+18y=1,741.02

-70x=--1,153.15‬

x=16.47

We now know that x (the price of the ribeye dinner) is roughly $16.47. What time being known, we can substitute $16.47 in the place of x in one of our original equations to solve for y (the value of the salmon dinner). Let use our first equation.

20x+18y=587.87

20(16.47)+18y=587.87

329.4‬0+18y=587.87

329.40-329.40+18y=587.87-329.40

18y=258.47‬

18y/18=258.47/18

y=14.36

We now know that the cost of a salmon dinner is $14.36. Therefore, to answer your original question, the cost of a ribeye dinner is 16.47 (x) and the cost of a salmon dinner is 14.36 (y). To confirm that the answers are, in fact, correct; simply substitute the value of the variables into EACH equation in the system of equations.

System of Equations:

20x+18y=587.87 ==> (20)(16.47)+(18)(14.36)=587.87=329.47+258.40=587.87 (confirmed)

30x+6y=580.34 ==> (30)(16.47)+(6)(14.36)=580.34=494.21+86.13=580.34 (confirmed)

Both values for x and y have been confirmed as valid when substituted in each equation. Therefore, you are done with the problem.