Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Scouting troops in Marshall County are putting on a crab feed to raise money for camp. They offer a complete crab dinner as well as a vegetarian option. One troop member sold tickets for $15$ crab meals and $17$ vegetarian meals, with total receipts of $\$890$ . Another sold tickets for $16$ crab meals and $17$ vegetarian meals, bringing in a total of $\$921$ . How much do the two types of tickets cost? $\newline$ A ticket for a crab meal costs $\$\_\_\_\_$ , and a ticket for a vegetarian meal costs $\$\_\_\_\_$ .

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Solve a system of equations using elimination: word problems

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

\newline

Scouting troops in Marshall County are putting on a crab feed to raise money for camp. They offer a complete crab dinner as well as a vegetarian option. One troop member sold tickets for

15

crab meals and

17

vegetarian meals, with total receipts of

\$890

. Another sold tickets for

16

crab meals and

17

vegetarian meals, bringing in a total of

\$921

. How much do the two types of tickets cost?

\newline

A ticket for a crab meal costs

\$\_\_\_\_

, and a ticket for a vegetarian meal costs

\$\_\_\_\_

Define Equations: Let's denote the cost of a crab meal ticket as $c$ and the cost of a vegetarian meal ticket as $v$ . We can then write two equations based on the information given: $\newline$ $1$ . For the first troop member: $15c + 17v = 890$ $\newline$ $2$ . For the second troop member: $16c + 17v = 921$
Eliminate Variable: To solve this system using elimination, we need to eliminate one of the variables. We can subtract the first equation from the second to eliminate $v$ : $\newline$ $(16c + 17v) - (15c + 17v) = 921 - 890$
Find Cost of Crab Meal: Simplifying the subtraction, we get: $\newline$ $16c - 15c + 17v - 17v = 921 - 890$ $\newline$ $c = 31$ $\newline$ So, a ticket for a crab meal costs $31$.
Substitute and Solve: Now that we know the cost of a crab meal ticket, we can substitute $ c = 31 $ into one of the original equations to find $ v $. Let's use the first equation:$\newline$$ 15(31) + 17v = 890 $
Calculate Vegetarian Meal Cost: Calculating the multiplication, we get:$\newline$$ 465 + 17v = 890 $$\newline$Now, we need to isolate $ v $ by subtracting $465$ from both sides of the equation:$\newline$$ 17v = 890 - 465 $
Calculate Vegetarian Meal Cost: Calculating the multiplication, we get:$\newline$$ 465 + 17v = 890 $$\newline$Now, we need to isolate $ v $ by subtracting $465$ from both sides of the equation:$\newline$$ 17v = 890 - 465 $Performing the subtraction, we find:$\newline$$ 17v = 425 $$\newline$To find $ v $, we divide both sides by $17$:$\newline$$ v = \frac{425}{17} $
Calculate Vegetarian Meal Cost: Calculating the multiplication, we get:$\newline$$ 465 + 17v = 890 $$\newline$Now, we need to isolate $ v $ by subtracting $465$ from both sides of the equation:$\newline$$ 17v = 890 - 465 $Performing the subtraction, we find:$\newline$$ 17v = 425 $$\newline$To find $ v $, we divide both sides by $17$:$\newline$$ v = \frac{425}{17} $Calculating the division, we get:$\newline$$ v = 25 $$\newline$So, a ticket for a vegetarian meal costs $25$ .