On Monday morning, Mr. Descartes asked his Algebra II students whether they had gone to the carnival in town over the weekend. "I did," said Kristen. "I went on the Teacups ride twice, the rollercoaster twice, and the Spinning Vortex once. I had a lot of fun for only $20."

"I went on those rides, too!" exclaimed Jorge. "Just once each on the Teacups and rollercoaster, but three times on the Spinning Vortex, all for $25."

"I did five Teacup rides," said Lakesha. "And two rollercoaster rides, and one Spinning Vortex. I spent $29."

Marc listed off the same rides. "Three Teacups for me, plus two rollercoasters, plus two rides on the Spinning Vortex, all for – "

"You don't need to say," interrupted Mr. Descartes. "I know how much you spent."

How much are the rides? How much did Marc spend

Cost of one Teacup =  $3

Cost of one Roller coaster =  $4

Cost of one Spinning Vortex = $6

Marc spent = 3x + 2y + 2z = $29

Step-by-step explanation:

Let

Cost of one Teacup = x

Cost of one Roller coaster = y

Cost of one Spinning Vortex = z

In Kristen case -

2x + 2y + z = $20

In Jorge case -

x + y + 3z = $25

In Lakesha case -

5x + 2y + z = $29

In Marc case -

3x + 2y + 2z = ?

Now, we have

2x + 2y + z = $20       (1)

x + y + 3z = $25         (2)

5x + 2y + z = $29      (3)

Now,

Subtract equation (1) from equation (3), we get

5x + 2y + z - ( 2x + 2y + z ) = $29 - 20

⇒5x + 2y + z - 2x - 2y - z  = $9

⇒3x = $9

⇒x = $3

Now,

Multiply be 2 in equation (2) , we get

2x + 2y + 6z = $50        (4)

Now,

Subtract equation (1) from equation (4), we get

2x + 2y + 6z - ( 2x + 2y + z ) = $50 - 20

⇒2x + 2y + 6z - 2x - 2y - z  = $30

⇒5z = $30

⇒z = $6

Now,

Put the values of x and z in equation (2) , we get

$3 + y + 3($6) = $25

⇒$3 + y + $18 = $25

⇒y + $21 = $25

⇒y = $25 - 21

⇒y = $4

∴ we get

Cost of one Teacup = x = $3

Cost of one Roller coaster = y = $4

Cost of one Spinning Vortex = z = $6

Marc spent = 3x + 2y + 2z = 3($3) + 2($4) + 2($6) = $9 + $8 + $12 = $29