jen butler has been pricing speed-pass train fares for a group trip to new york. three adults and four children must pay $83. two adults and three children must pay $59. find the price of the adults ticket and the price of a child’s ticket.

Let's solve this step by step.

Step 1: Assign variables

Let's assign variables to the unknowns:

Let the price of an adult's ticket be represented by "A"
Let the price of a child's ticket be represented by "C"

Step 2: Set up equations

From the given information, we can set up two equations:

Equation 1: 3A + 4C = 83 (Three adults and four children must pay $83)
Equation 2: 2A + 3C = 59 (Two adults and three children must pay $59)

Step 3: Solve the equations

To solve these equations, we can use the method of substitution or elimination. Let's use the method of elimination.

Multiply equation 2 by 2 to make the coefficients of "A" in both equations the same:

Equation 1: 3A + 4C = 83
Equation 2: 4A + 6C = 118

Now, subtract equation 1 from equation 2:

Equation 2 - Equation 1: (4A + 6C) - (3A + 4C) = 118 - 83
4A + 6C - 3A - 4C = 35
A + 2C = 35 --> Equation 3

Step 4: Solve for "A" in equation 3

Now, we have a new equation:

A + 2C = 35

We can solve this equation for "A" in terms of "C":

A = 35 - 2C --> Equation 4

Step 5: Substitute the value of "A" in equation 4 back into equation 1 or 2

Let's substitute the value of "A" in equation 4 back into equation 1:

3(35 - 2C) + 4C = 83
105 - 6C + 4C = 83
105 - 2C = 83
-2C = 83 - 105
-2C = -22
C = -22 / -2
C = 11

Step 6: Find the value of "A" using the found value of "C"

Substitute the value of "C" (which is 11) back into equation 4:

A = 35 - 2(11)
A = 35 - 22
A = 13

So, the price of an adult's ticket is $13 and the price of a child's ticket is $11.

To solve this problem, we can set up a system of equations to represent the given information.

Let's assume the price of an adult's ticket is 'a', and the price of a child's ticket is 'c'.

From the given information, we have two equations:

Equation 1:
3a + 4c = 83 (Total price for three adults and four children)

Equation 2:
2a + 3c = 59 (Total price for two adults and three children)

Now, we can solve this system of equations to find the values of 'a' and 'c'.

One way to solve the system of equations is by elimination. Multiply Equation 1 by 2 and Equation 2 by 3 to make the 'a' coefficients equal:

Equation 1 (multiplied by 2):
6a + 8c = 166

Equation 2 (multiplied by 3):
6a + 9c = 177

Now, subtract Equation 2 from Equation 1 to eliminate 'a':

6a + 8c - (6a + 9c) = 166 - 177
6a + 8c - 6a - 9c = -11
-c = -11

Divide both sides of the equation by -1 to solve for 'c':

c = 11

Now that we know the price of a child's ticket is $11, we can substitute this value into either Equation 1 or Equation 2 to find the price of an adult's ticket.

Using Equation 2:
2a + 3c = 59
2a + 3(11) = 59
2a + 33 = 59
2a = 59 - 33
2a = 26
a = 13

Therefore, the price of an adult's ticket is $13, and the price of a child's ticket is $11.

3a+4c = 83

2a+3c = 59
now solve as usual