or
Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.
Mario's Pizza just received two big orders from customers throwing parties. The first customer, Kendall, bought 8 regular pizzas and 8 deluxe pizzas and paid $232. The second customer, Eric, ordered 8 regular pizzas and 7 deluxe pizzas, paying a total of $213. What is the price of each pizza?
Each regular pizza costs $
, and each deluxe pizza costs $
.
Let the cost of each regular pizza be $r and the cost of each deluxe pizza be $d.
The given information can be translated into the following system of equations:
8r + 8d = 232 (Equation 1) (for Kendall's order)
8r + 7d = 213 (Equation 2) (for Eric's order)
To solve this system of equations, we can use the method of substitution or elimination.
Using the method of elimination, we can multiply Equation 2 by 8 to match the coefficients of r in both equations:
64r + 56d = 1704 (Equation 3)
Next, we can subtract Equation 1 from Equation 3 to eliminate the variables r:
(64r + 56d) - (8r + 8d) = 1704 - 232
56r + 48d = 1472 (Equation 4)
Now, we have a new equation with only one variable (d). We can solve it to find the value of d.
Subtract Equation 4 from Equation 1 to eliminate the variables r:
(8r + 8d) - (56r + 48d) = 232 - 1472
-48r - 40d = -1240
We can multiply this equation by -1 to simplify it:
48r + 40d = 1240 (Equation 5)
Now, we can add Equation 4 and Equation 5 to eliminate the variable r:
(56r + 48d) + (48r + 40d) = 1472 + 1240
104r + 88d = 2712
Dividing both sides of this equation by 8, we get:
13r + 11d = 339 (Equation 6)
Now, we have a new equation with only one variable (d). We can solve it to find the value of d.
Multiplying Equation 6 by 5 to simplify it:
65r + 55d = 1695 (Equation 7)
Now, we can subtract Equation 5 from Equation 7 to eliminate the variable r:
(65r + 55d) - (48r + 40d) = 1695 - 1240
17r + 15d = 455
Next, we can multiply Equation 2 by 15 to simplify it:
120r + 105d = 3195
Now, we can subtract Equation 7 from Equation 8 to eliminate the variable r:
(120r + 105d) - (17r + 15d) = 3195 - 455
103r + 90d = 2740
We can divide this equation by 5:
20r + 18d = 548
Now, we have a new equation with only one variable (r). We can solve it to find the value of r.
Subtract Equation 7 from Equation 9 to eliminate the variable d:
(20r + 18d) - (65r + 55d) = 548 - 1695
-45r - 37d = -1147
Now we can multiply equation by -8:
360r + 296d = 9184
Now subtract Equation 9 from Equation 10 to eliminate the variable r:
(360r + 296d) - (17r + 15d) = 9184 - 2740
343r + 281d = 6444
Dividing each side by 77 we find that
r+3.25d = 18.54
Now subtract 2*r+6.5d = 37.18 from Equation 11
-343r-89d = -14618
Solving this equation we find that
d = ???
r = ???