Answer :
To formulate the system of linear equations based on the given conditions, let's consider each piece of information step-by-step.
### Given Information:
1. Total Number of Tickets Sold:
- There are three types of tickets: [tex]$8, $[/tex]10, and [tex]$12.
- The total number of tickets sold is 400.
- Let \( x \) represent the number of $[/tex]8 tickets.
- Let [tex]\( y \)[/tex] represent the number of [tex]$10 tickets.
- Let \( z \) represent the number of $[/tex]12 tickets.
So, we have the equation:
[tex]\[
x + y + z = 400
\][/tex]
2. Total Income from Ticket Sales:
- The number of [tex]$8 tickets multiplied by 8 gives the income from the $[/tex]8 tickets.
- The number of [tex]$10 tickets multiplied by 10 gives the income from the $[/tex]10 tickets.
- The number of [tex]$12 tickets multiplied by 12 gives the income from the $[/tex]12 tickets.
- The total income from all ticket sales is [tex]$3700.
So, we have the equation:
\[
8x + 10y + 12z = 3700
\]
3. Relationship Between the Number of Tickets Sold:
- The combined number of $[/tex]8 and [tex]$10 tickets sold (i.e., \( x + y \)) is 7 times the number of $[/tex]12 tickets sold (i.e., [tex]\( z \)[/tex]).
So, we have the equation:
[tex]\[
x + y = 7z
\][/tex]
### Formulating the System of Linear Equations:
Hence, the system of linear equations based on the given conditions is:
1. [tex]\( x + y + z = 400 \)[/tex]
2. [tex]\( 8x + 10y + 12z = 3700 \)[/tex]
3. [tex]\( x + y = 7z \)[/tex]
### Multiple Choice Identification:
Comparing the formulated equations with the provided choices, we find:
Option d:
[tex]\[
\begin{array}{l}
x + y + z = 400 \\
8x + 10y + 12z = 3700 \\
x + y = 7z
\end{array}
\][/tex]
This matches perfectly with our formulated system of equations.
Thus, the correct choice is:
d
[tex]\[
\begin{array}{l}
x + y + z = 400 \\
8x + 10y + 12z = 3700 \\
x + y = 7z
\end{array}
\][/tex]
### Given Information:
1. Total Number of Tickets Sold:
- There are three types of tickets: [tex]$8, $[/tex]10, and [tex]$12.
- The total number of tickets sold is 400.
- Let \( x \) represent the number of $[/tex]8 tickets.
- Let [tex]\( y \)[/tex] represent the number of [tex]$10 tickets.
- Let \( z \) represent the number of $[/tex]12 tickets.
So, we have the equation:
[tex]\[
x + y + z = 400
\][/tex]
2. Total Income from Ticket Sales:
- The number of [tex]$8 tickets multiplied by 8 gives the income from the $[/tex]8 tickets.
- The number of [tex]$10 tickets multiplied by 10 gives the income from the $[/tex]10 tickets.
- The number of [tex]$12 tickets multiplied by 12 gives the income from the $[/tex]12 tickets.
- The total income from all ticket sales is [tex]$3700.
So, we have the equation:
\[
8x + 10y + 12z = 3700
\]
3. Relationship Between the Number of Tickets Sold:
- The combined number of $[/tex]8 and [tex]$10 tickets sold (i.e., \( x + y \)) is 7 times the number of $[/tex]12 tickets sold (i.e., [tex]\( z \)[/tex]).
So, we have the equation:
[tex]\[
x + y = 7z
\][/tex]
### Formulating the System of Linear Equations:
Hence, the system of linear equations based on the given conditions is:
1. [tex]\( x + y + z = 400 \)[/tex]
2. [tex]\( 8x + 10y + 12z = 3700 \)[/tex]
3. [tex]\( x + y = 7z \)[/tex]
### Multiple Choice Identification:
Comparing the formulated equations with the provided choices, we find:
Option d:
[tex]\[
\begin{array}{l}
x + y + z = 400 \\
8x + 10y + 12z = 3700 \\
x + y = 7z
\end{array}
\][/tex]
This matches perfectly with our formulated system of equations.
Thus, the correct choice is:
d
[tex]\[
\begin{array}{l}
x + y + z = 400 \\
8x + 10y + 12z = 3700 \\
x + y = 7z
\end{array}
\][/tex]
