Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Mario's Pizza just received two big orders from customers throwing parties. The first customer, Melissa, bought $4$ regular pizzas and $3$ deluxe pizzas and paid $\$111$ . The second customer, Pablo, ordered $3$ regular pizzas and $10$ deluxe pizzas, paying a total of $\$246$ . What is the price of each pizza? $\newline$ Each regular pizza costs $\$$ _, and each deluxe pizza costs $\$$ _.

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

Mario's Pizza just received two big orders from customers throwing parties. The first customer, Melissa, bought

4

regular pizzas and

3

deluxe pizzas and paid

\$111

. The second customer, Pablo, ordered

3

regular pizzas and

10

deluxe pizzas, paying a total of

\$246

. What is the price of each pizza?

\newline

Each regular pizza costs

\$

_____, and each deluxe pizza costs

\$

_____.

Define Prices: Let's denote the price of a regular pizza as $r$ and the price of a deluxe pizza as $d$ . Melissa's order gives us the first equation: $4r + 3d = 111$ .
Form Equations: Pablo's order gives us the second equation: $3r + 10d = 246$ .
Elimination Method: We now have a system of equations to solve: $\newline$ $4r + 3d = 111$ $\newline$ $3r + 10d = 246$ $\newline$ We can use either substitution or elimination to solve this system. Let's use the elimination method.
Eliminate Variable: To eliminate one of the variables, we can multiply the first equation by $10$ and the second equation by $3$ to make the coefficients of $d$ the same: $\newline$ $(4r + 3d) \times 10 = 111 \times 10$ $\newline$ $(3r + 10d) \times 3 = 246 \times 3$ $\newline$ This gives us: $\newline$ $40r + 30d = 1110$ $\newline$ $9r + 30d = 738$
Solve for $r$ : Now we subtract the second equation from the first to eliminate $d$ :
$(40r + 30d) - (9r + 30d) = 1110 - 738$
$40r - 9r = 1110 - 738$
$31r = 372$
Substitute and Solve: Divide both sides by $31$ to solve for $r$ : $\frac{31r}{31} = \frac{372}{31}$ $r = 12$
Find Value of d: Now that we have the value for $r$ , we can substitute it back into one of the original equations to solve for $d$ . Let's use the first equation: $\newline$ $4(12) + 3d = 111$ $\newline$ $48 + 3d = 111$
Find Value of d: Now that we have the value for $r$ , we can substitute it back into one of the original equations to solve for $d$ . Let's use the first equation: $\newline$ $4(12) + 3d = 111$ $\newline$ $48 + 3d = 111$ Subtract $48$ from both sides to solve for $d$ : $\newline$ $3d = 111 - 48$ $\newline$ $3d = 63$
Find Value of d: Now that we have the value for $r$ , we can substitute it back into one of the original equations to solve for $d$ . Let's use the first equation: $\newline$ $4(12) + 3d = 111$ $\newline$ $48 + 3d = 111$ Subtract $48$ from both sides to solve for $d$ : $\newline$ $3d = 111 - 48$ $\newline$ $3d = 63$ Divide both sides by $3$ to find the value of $d$ : $\newline$ $\frac{3d}{3} = \frac{63}{3}$ $\newline$ $d = 21$