How much time does Martin spend riding the Ferris wheel today?

Understand the Problem

The question is asking how much time Martin spends riding the Ferris wheel today, given a system of equations that describes the relationship between the time spent on the Ferris wheel and the roller coaster.

Answer

Martin spends $2$ minutes riding the Ferris wheel today.

Steps to Solve

Identify the Variables Let ( x ) represent the time it takes to ride the roller coaster once in minutes, and ( y ) represent the time it takes to ride the Ferris wheel once in minutes.
Set Up the Equations From the problem, we have the following system of equations: $$ x + y = 14 $$ $$ 6x + 9y = 90 $$
Solve the First Equation for ( y ) We can express ( y ) in terms of ( x ): $$ y = 14 - x $$
Substitute ( y ) in the Second Equation Now, we substitute ( y ) in the second equation: $$ 6x + 9(14 - x) = 90 $$
Simplify the Second Equation Distributing the 9: $$ 6x + 126 - 9x = 90 $$
Combine Like Terms Combine the ( x ) terms: $$ -3x + 126 = 90 $$
Isolate ( x ) Subtract 126 from both sides: $$ -3x = 90 - 126 $$ $$ -3x = -36 $$
Solve for ( x ) Divide both sides by -3: $$ x = 12 $$
Substitute ( x ) Back to Find ( y ) Now that we have ( x ), substitute it back into the equation for ( y ): $$ y = 14 - 12 $$ $$ y = 2 $$
Conclusion The time spent riding the Ferris wheel today is ( y ).

Martin spends $2$ minutes riding the Ferris wheel today.

More Information

Thus, Martin rides the Ferris wheel for a total of 2 minutes, while the roller coaster takes him 12 minutes. Together, these times match the total time spent riding both attractions.

Tips

A common mistake is to misinterpret the equations. Always ensure that the definitions of ( x ) and ( y ) are consistent with the problem statement.
Another error is forgetting to substitute correctly, which can lead to incorrect values for ( x ) or ( y ).

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