A group of 2 adults and 4 children spent $38 on tickets to a museum. A group of 3 adults and 3 children spent $40.50 on tickets to the museum. Based on this information, how much is an adult ticket, and how much is a child ticket?
[tex]\begin{array}{l}
2 a+4 c=38.00 \\
3 a+3 c=40.50
\end{array}[/tex]
What values of [tex]$a$[/tex] and [tex]$c$[/tex] represent the solution to the system?
[tex]$a=\square$[/tex]
[tex]$c=\square$[/tex]
Answer (1)
Set up a system of two linear equations: 2 a + 4 c = 38 and 3 a + 3 c = 40.50 .
Use the elimination method by multiplying the equations to equate the coefficients of a .
Subtract the equations to eliminate a and solve for c : c = 6 33 = 5.50 .
Substitute the value of c back into one of the original equations to solve for a : a = 8 . The final answer is a = 8 , c = 5.5
Explanation
Problem Analysis Let's analyze the problem. We have a system of two linear equations with two unknowns, the price of an adult ticket ( a ) and the price of a child ticket ( c ). Our goal is to find the values of a and c that satisfy both equations.
Stating the Equations The given equations are: 2 a + 4 c = 38 3 a + 3 c = 40.50
Elimination Method - Multiplication To solve this system, we can use the method of elimination. First, multiply the first equation by 3 and the second equation by 2 to make the coefficients of a the same: 3 ( 2 a + 4 c ) = 3 ( 38 ) ⇒ 6 a + 12 c = 114 2 ( 3 a + 3 c ) = 2 ( 40.50 ) ⇒ 6 a + 6 c = 81
Elimination Method - Subtraction Now, subtract the second equation from the first equation to eliminate a :
( 6 a + 12 c ) − ( 6 a + 6 c ) = 114 − 81 6 c = 33
Solving for c Solve for c :
c = 6 33 = 5.50
Substituting c into Equation 1 Now that we have the value of c , we can substitute it back into one of the original equations to solve for a . Let's use the first equation: 2 a + 4 ( 5.50 ) = 38 2 a + 22 = 38
Solving for a Solve for a :
2 a = 38 − 22 2 a = 16 a = 2 16 = 8
Final Answer So, the price of an adult ticket is $8 and the price of a child ticket is $5.50 .
Examples
Understanding systems of equations is incredibly useful in everyday life. For instance, imagine you're planning a balanced diet. You know the nutritional content of two different foods and want to figure out how much of each to eat to meet specific calorie and nutrient goals. By setting up a system of equations, you can determine the precise quantities needed to achieve your dietary targets, ensuring you get the right balance of nutrients without exceeding your calorie limits. This is just one example of how mathematical problem-solving can directly impact your daily decisions and health.
