Task $4$ : Write and solve a system of equations to model each situation. $\newline$ $1$ . Two pizzas and four sandwiches cost $\$ 62$ . Four pizzas and ten sandwiches cost $\$ 140$ . How much does each pizza and sandwich cost? $\newline$ $\int 2 x+4 y=62$ $\newline$ $\begin{array}{l} 4 x+10 y=140 \\ (2 x \cdot 2)+(4 y \cdot 2)+(62 \cdot 2) \\ 7 \\ 4 x+8 y=124 \\ 4 x+10 y=140 \\ -2 y=-115 \end{array}$ $\newline$ $(2 x \cdot 2)$ $\newline$ $2$ . 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?

Full solution

Q. Task

4

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

\newline

1

. Two pizzas and four sandwiches cost

\$ 62

. Four pizzas and ten sandwiches cost

\$ 140

. How much does each pizza and sandwich cost?

\newline

\int 2 x+4 y=62

\newline

\begin{array}{l} 4 x+10 y=140 \\ (2 x \cdot 2)+(4 y \cdot 2)+(62 \cdot 2) \\ 7 \\ 4 x+8 y=124 \\ 4 x+10 y=140 \\ -2 y=-115 \end{array}

\newline

(2 x \cdot 2)

\newline

2

. 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?

Multiply by $2$ : Now, let's multiply Equation $1$ by $2$ to help us eliminate one of the variables when we subtract the equations from each other. $\newline$ $(2x + 4y) \times 2 = 62 \times 2$ $\newline$ $4x + 8y = 124$ (Equation $3$ )
Subtract to eliminate $x$ : Next, we will subtract Equation $3$ from Equation $2$ to eliminate the $x$ variable and solve for $y$ . $4x + 10y - (4x + 8y) = 140 - 124$ $4x + 10y - 4x - 8y = 16$ $2y = 16$
Solve for y: Now, we divide both sides by $2$ to find the value of y. $\newline$ $\frac{2y}{2} = \frac{16}{2}$ $\newline$ $y = 8$ $\newline$ So, each sandwich costs $\$8$ .
Substitute back into Equation $1$ : With the value of $y$ known, we can substitute it back into Equation $1$ to find the value of $x$ . $2x + 4(8) = 62$ $2x + 32 = 62$
Solve for x: Subtract $32$ from both sides to solve for $x$ . $\newline$ $2x + 32 - 32 = 62 - 32$ $\newline$ $2x = 30$
Final solution: Finally, divide both sides by $2$ to find the value of $x$ . $\frac{2x}{2} = \frac{30}{2}$ $x = 15$ So, each pizza costs $\$15$ .