Answer :
To solve for the number of guests that were children (denoted as [tex]\( x \)[/tex]) and the number of guests that were adults (denoted as [tex]\( y \)[/tex]), we analyze the given system of equations:
[tex]\[ \begin{aligned} 1. \quad 12x + 25y &= 9014 \\ 2. \quad x + y &= 450 \end{aligned} \][/tex]
First, we solve the second equation for one of the variables. Let's solve for [tex]\( y \)[/tex]:
[tex]\[ y = 450 - x \][/tex]
Next, we substitute [tex]\( y = 450 - x \)[/tex] into the first equation:
[tex]\[ 12x + 25(450 - x) = 9014 \][/tex]
Expanding and simplifying the equation:
[tex]\[ 12x + 11250 - 25x = 9014 \][/tex]
Combine like terms:
[tex]\[ -13x + 11250 = 9014 \][/tex]
Subtract 11250 from both sides of the equation:
[tex]\[ -13x = 9014 - 11250 \][/tex]
[tex]\[ -13x = -2236 \][/tex]
Divide both sides by -13:
[tex]\[ x = \frac{-2236}{-13} \][/tex]
[tex]\[ x = 172 \][/tex]
So, the number of children is [tex]\( 172 \)[/tex].
To find the number of adults, we substitute [tex]\( x = 172 \)[/tex] back into the second equation:
[tex]\[ x + y = 450 \][/tex]
[tex]\[ 172 + y = 450 \][/tex]
Subtract 172 from both sides:
[tex]\[ y = 450 - 172 \][/tex]
[tex]\[ y = 278 \][/tex]
So, the number of adults is [tex]\( 278 \)[/tex].
Therefore, the approximate number of guests that were children for that day is 172, and the approximate number of guests that were adults for that day is 278.
[tex]\[ \begin{aligned} 1. \quad 12x + 25y &= 9014 \\ 2. \quad x + y &= 450 \end{aligned} \][/tex]
First, we solve the second equation for one of the variables. Let's solve for [tex]\( y \)[/tex]:
[tex]\[ y = 450 - x \][/tex]
Next, we substitute [tex]\( y = 450 - x \)[/tex] into the first equation:
[tex]\[ 12x + 25(450 - x) = 9014 \][/tex]
Expanding and simplifying the equation:
[tex]\[ 12x + 11250 - 25x = 9014 \][/tex]
Combine like terms:
[tex]\[ -13x + 11250 = 9014 \][/tex]
Subtract 11250 from both sides of the equation:
[tex]\[ -13x = 9014 - 11250 \][/tex]
[tex]\[ -13x = -2236 \][/tex]
Divide both sides by -13:
[tex]\[ x = \frac{-2236}{-13} \][/tex]
[tex]\[ x = 172 \][/tex]
So, the number of children is [tex]\( 172 \)[/tex].
To find the number of adults, we substitute [tex]\( x = 172 \)[/tex] back into the second equation:
[tex]\[ x + y = 450 \][/tex]
[tex]\[ 172 + y = 450 \][/tex]
Subtract 172 from both sides:
[tex]\[ y = 450 - 172 \][/tex]
[tex]\[ y = 278 \][/tex]
So, the number of adults is [tex]\( 278 \)[/tex].
Therefore, the approximate number of guests that were children for that day is 172, and the approximate number of guests that were adults for that day is 278.