Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. $\newline$ Andy loves riding Ferris wheels and roller coasters. While visiting the Morgan County Fair, he first went on the Ferris wheel $1$ time, using a total of $2$ tickets. Then, after taking a break and having a snack, Andy went on the Ferris wheel $2$ times and the roller coaster $4$ times, using a total of $20$ tickets. How many tickets does it take to ride each attraction? $\newline$ It takes $\_\_\_\_$ tickets to ride the Ferris wheel, and $\_\_\_\_$ tickets to ride the roller coaster.

Full solution

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

\newline

Andy loves riding Ferris wheels and roller coasters. While visiting the Morgan County Fair, he first went on the Ferris wheel

1

time, using a total of

2

tickets. Then, after taking a break and having a snack, Andy went on the Ferris wheel

2

times and the roller coaster

4

times, using a total of

20

tickets. How many tickets does it take to ride each attraction?

\newline

It takes

\_\_\_\_

tickets to ride the Ferris wheel, and

\_\_\_\_

tickets to ride the roller coaster.

Define Variables: Let $x$ be the number of tickets for the Ferris wheel and $y$ be the number of tickets for the roller coaster. Andy's first ride gives the equation $1x + 0y = 2$ .
Form Equations: Andy's second set of rides gives the equation $2x + 4y = 20$ .
Create Augmented Matrix: Write the system of equations as an augmented matrix: $\left[\begin{array}{ccc} 1 & 0 & 2 \ 2 & 4 & 20 \end{array}\right]$ .
Eliminate Variable $x$ : Multiply the first row by $-2$ and add it to the second row to eliminate $x$ from the second equation: new second row is $[0, 4, 16]$ .
Solve for y: Divide the second row by $4$ to solve for $y$ : $y = 4$ .
Solve for $x$ : Substitute $y = 4$ into the first equation to solve for $x$ : $1x + 0(4) = 2$ , so $x = 2$ .
Check Solution: Check the solution by substituting $x$ and $y$ into the second equation: $2(2) + 4(4) = 4 + 16 = 20$ , which is correct.