Write and solve a system of equations to model each situation.

Two pizzas and four sandwiches cost $62. Four pizzas and ten sandwiches cost $140. How much does each pizza and sandwich cost?
At a clothing store, 3 shirts and 8 hats cost $65. The cost for 2 shirts and 2 hats is $30. How much does each shirt and hat cost?

Question

dineshafoster1990 dineshafoster1990

05-11-2026
Mathematics

Answered

Write and solve a system of equations to model each situation.

Two pizzas and four sandwiches cost $62. Four pizzas and ten sandwiches cost $140. How much does each pizza and sandwich cost?
At a clothing store, 3 shirts and 8 hats cost $65. The cost for 2 shirts and 2 hats is $30. How much does each shirt and hat cost?

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-05T17:57:05-05:00

For the first system (2 pizzas and 4 sandwiches for $\$62$, 4 pizzas and 10 sandwiches for $\$140$), we solve to find that the price of a pizza is $\$15$ and a sandwich is $\$8$.
For the second system (3 shirts and 8 hats for $\$65$, 2 shirts and 2 hats for $\$30$), we solve to find that the price of a shirt is $\$11$ and a hat is $\$4$.
Final results: Pizza: $\boxed{\$15}$, Sandwich: $\boxed{\$8}$, Shirt: $\boxed{\$11}$, Hat: $\boxed{\$4}$.

Explanation

Analysis of the Problems

We are presented with two separate problems involving systems of linear equations. In the first problem (Question 26), we need to find the price of a pizza $P$ and a sandwich $S$. We are given that 2 pizzas and 4 sandwiches cost $\$62$, leading to the equation $2P + 4S = 62$. We also know that 4 pizzas and 10 sandwiches cost $\$140$, giving the equation $4P + 10S = 140$. In the second problem (Question 27), we need to find the price of a shirt $X$ and a hat $H$. We are given that 3 shirts and 8 hats cost $\$65$, resulting in the equation $3X + 8H = 65$, and 2 shirts and 2 hats cost $\$30$, resulting in the equation $2X + 2H = 30$. Our goal is to solve these systems of linear equations to determine the individual costs for each item.

Solving the First System of Equations

To solve the first system for $P$ and $S$:

Start with the equations:

\[2P + 4S = 62\]
\[4P + 10S = 140\]

Simplify the first equation by dividing by 2:

\[P + 2S = 31 \implies P = 31 - 2S\]

Substitute this expression for $P$ into the second equation:

\[4(31 - 2S) + 10S = 140\]
\[124 - 8S + 10S = 140\]
\[2S = 16 \implies S = 8\]

Now, find $P$ by substituting $S = 8$ back into the expression for $P$:

\[P = 31 - 2(8) = 31 - 16 = 15\]
Thus, a pizza costs $\$15$ and a sandwich costs $\$8$.

Solving the Second System of Equations

To solve the second system for $X$ and $H$:

Start with the equations:

\[3X + 8H = 65\]
\[2X + 2H = 30\]

Simplify the second equation by dividing by 2:

\[X + H = 15 \implies X = 15 - H\]

Substitute this expression for $X$ into the first equation:

\[3(15 - H) + 8H = 65\]
\[45 - 3H + 8H = 65\]
\[5H = 20 \implies H = 4\]

Now, find $X$ by substituting $H = 4$ back into the expression for $X$:

\[X = 15 - 4 = 11\]
Thus, a shirt costs $\$11$ and a hat costs $\$4$.

Conclusion and Results

The prices determined from the systems are as follows:

For the pizza and sandwich: $P = \$15$ and $S = \$8$.
For the shirt and hat: $X = \$11$ and $H = \$4$.

We have successfully solved both systems using the substitution method. Each calculation is consistent with the given constraints. Therefore, the costs are: pizza $\$15$, sandwich $\$8$, shirt $\$11$, and hat $\$4$.

Examples

Systems of linear equations are incredibly useful in everyday shopping scenarios. Imagine you are organizing an event and need to budget for catering. If you know the total cost of two different menu combinations, you can use algebra to figure out the individual prices of the items, helping you compare prices between vendors or decide how many of each item you can afford. Similarly, when shopping for clothing sets or grocery bundles, understanding the cost per unit allows you to make informed decisions about your spending and identify the best deals available.