Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ At a community barbecue, Mrs. Davidson and Mr. Ayala are buying dinner for their families. Mrs. Davidson purchases $2$ hot dog meals and $3$ hamburger meals, paying a total of $\$39$ . Mr. Ayala buys $3$ hot dog meals and $3$ hamburger meals, spending $\$45$ in all. How much do the meals cost? $\newline$ Hot dog meals cost $\$$ _ each, and hamburger meals cost $\$$ _ each.

Full solution

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

\newline

At a community barbecue, Mrs. Davidson and Mr. Ayala are buying dinner for their families. Mrs. Davidson purchases

2

hot dog meals and

3

hamburger meals, paying a total of

\$39

. Mr. Ayala buys

3

hot dog meals and

3

hamburger meals, spending

\$45

in all. How much do the meals cost?

\newline

Hot dog meals cost

\$

_____ each, and hamburger meals cost

\$

_____ each.

Define Prices: Let's denote the price of each hot dog meal as $h$ and the price of each hamburger meal as $d$ . Mrs. Davidson purchased $2$ hot dog meals and $3$ hamburger meals for a total of $\$$ $39$ . This gives us the equation $2h + 3d = 39$ .
Mrs. Davidson's Purchase: Mr. Ayala purchased $3$ hot dog meals and $3$ hamburger meals for a total of $\$45$ . This gives us the equation $3h + 3d = 45$ .
Eliminate Variable: We now have a system of two equations. We need to eliminate one of the variables, $h$ or $d$ . We choose to eliminate $h$ because its coefficients are the same in both equations.
Adjust Equations: To eliminate $h$ , we multiply the first equation by $-\frac{3}{2}$ to match the coefficient of $h$ in the second equation. This gives us the new equation $-3h - 4.5d = -58.5$ .
Solve for $d$ : We now add the new first equation to the second equation to eliminate $h$ . This gives us $-1.5d = -13.5$ , or $d = 9$ after dividing both sides by $-1.5$ .
Solve for $h$ : We substitute $d = 9$ into the first equation and solve for $h$ . This gives us $2h + 27 = 39$ , or $2h = 12$ after subtracting $27$ from both sides, and finally $h = 6$ after dividing both sides by $2$ .