Step 1: Assign variable for the unknown that we need to find.
Let x be the charge/person who are under age 12
Let y be the charge/person who are over age 12, and
Let z be the charge/person who are adults.
Step 2: Based on the information given write mathematical equation
Statement 1: "A family with 2 kids under age 12, 3 kids over age 12, and 3 adults pays $174 for entrance to the park"
For Under Age 12, 1 kid charge $x, so 2 kids under age 12 charge will be $2x.
For Over Age 12, 1 kid charge $y, so 3 Kids over age 12 charge will be $3y.
For Adults, 1 Adult charge $z, so 3 Adults charge will be $3z
Based on the above statement which mentions the number of person in the family and total cost of $174, we can write our 1st equation as
2x + 3y + 3z = 174 ⇒ 1st Equation
Statement 2: "A mother and father bring their 4 kids under age 12 at a cost of $122"
4 kids under 12 will be charged $4x, & 2 Adults (Mother and father) will be charged $2z, and total cost is given as $122.
Based on the above statement 2, we can write the 2nd equation as
4x + 2z = 122 ⇒ 2nd Equation
Statement 3: A grandmother who brings her teenage granddaughter to the park pays $46.
Grandmother will be considered an adult with teenage granddaughter who need to be considered as kid over age 12.
So we can write the third equation as
y + z = 46 ⇒ 3rd equation.
Step 3: Solve the three equation using substitution method
2x + 3y + 3z = 174 ⇒ 1st Equation
4x + 2z = 122 ⇒ 2nd Equation
y + z = 46 ⇒ 3rd equation
Factoring 3 in the left side of the last two terms of 1st equation we get...
2x + 3(y + z) = 174
Then substituting 46 for y + z we get
2x + 3(46) = 174
2x + 138 = 174 { Subtracting 138 on both sides }
2x +138 - 138 = 174 - 138 {Combining like terms on either side of the equation}
2x = 36 {Dividing 2 on both sides}
2x/2 = 36/2 {Simplifying each fraction}
<em>x = 18</em>
Substituting 18 for x in 2nd equation, 4x + 2z = 122
4(18) + 2z = 122
72 + 2z = 122 {Subtracting 72 on both sides of the equation}
72 + 2z - 72 = 122 - 72 {Combining like terms on both sides of the equation}
2z = 50 {Dividing by 2 on both sides}
2z/2 = 50/2 {Simplifying fraction on both sides of equation}
<em>z = 25</em>
Substituting 25 for z in the 3rd equation y + z = 46
y + 25 = 46 { Subtracting 25 on both sides of the equation }
y + 25 - 25 = 46 - 25 {Combining like terms on either side of the equation}
<em>y = 21</em>
Conclusion:
The cost for an Kid who are under age 12 to enter the theme park = $18
The cost for an Kid who are over age 12 to enter the theme park = $21
<u><em>The cost for an adult to enter the theme park = $25</em></u>
, for n=1 becomes 
.
with 
, we see that