Answer :
To determine which tables represent viable solutions for the equation [tex]\( y = 5x \)[/tex], where [tex]\( x \)[/tex] is the number of tickets sold for the school play and [tex]\( y \)[/tex] is the amount of money collected, we will check each table to see if it satisfies the equation.
We do this by examining each pair [tex]\((x, y)\)[/tex] in the table to verify if [tex]\( y = 5x \)[/tex].
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } (x) & \text{Money Collected } (y) \\ \hline-100 & -500 \\ \hline-25 & -125 \\ \hline 40 & 250 \\ \hline 600 & 3,000 \\ \hline \end{array} \][/tex]
- For [tex]\( x = -100 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times -100 = -500 \)[/tex]. (True)
- For [tex]\( x = -25 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times -25 = -125 \)[/tex]. (True)
- For [tex]\( x = 40 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 40 = 250 \)[/tex]. (True)
- For [tex]\( x = 600 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 600 = 3000 \)[/tex]. (True)
All values in Table 1 satisfy the equation [tex]\( y = 5x \)[/tex].
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } (x) & \text{Money Collected } (y) \\ \hline-20 & -100 \\ \hline 20 & 100 \\ \hline 100 & 500 \\ \hline 109 & 545 \\ \hline \end{array} \][/tex]
- For [tex]\( x = -20 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times -20 = -100 \)[/tex]. (True)
- For [tex]\( x = 20 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 20 = 100 \)[/tex]. (True)
- For [tex]\( x = 100 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 100 = 500 \)[/tex]. (True)
- For [tex]\( x = 109 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 109 = 545 \)[/tex]. (True)
All values in Table 2 satisfy the equation [tex]\( y = 5x \)[/tex].
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } ( x ) & \text{Money Collected } ( y ) \\ \hline 0 & 0 \\ \hline 10 & 50 \\ \hline 51 & 255 \\ \hline 400 & 2000 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 0 = 0 \)[/tex]. (True)
- For [tex]\( x = 10 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 10 = 50 \)[/tex]. (True)
- For [tex]\( x = 51 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 51 = 255 \)[/tex]. (True)
- For [tex]\( x = 400 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 400 = 2000 \)[/tex]. (True)
All values in Table 3 satisfy the equation [tex]\( y = 5x \)[/tex].
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } (x) & \text{Money Collected } (y) \\ \hline 5 & 25 \\ \hline 65 & 350 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 5 = 25 \)[/tex]. (True)
- For [tex]\( x = 65 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 65 = 325 \)[/tex]. (False)
In Table 4, only the first pair satisfies the equation [tex]\( y = 5x \)[/tex]. The second pair does not.
### Conclusion:
The tables that represent viable solutions for [tex]\( y = 5x \)[/tex] are:
- [1]
- [2]
- [3]
Therefore, Table 1, Table 2, and Table 3 are the viable solutions for the given equation.
We do this by examining each pair [tex]\((x, y)\)[/tex] in the table to verify if [tex]\( y = 5x \)[/tex].
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } (x) & \text{Money Collected } (y) \\ \hline-100 & -500 \\ \hline-25 & -125 \\ \hline 40 & 250 \\ \hline 600 & 3,000 \\ \hline \end{array} \][/tex]
- For [tex]\( x = -100 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times -100 = -500 \)[/tex]. (True)
- For [tex]\( x = -25 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times -25 = -125 \)[/tex]. (True)
- For [tex]\( x = 40 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 40 = 250 \)[/tex]. (True)
- For [tex]\( x = 600 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 600 = 3000 \)[/tex]. (True)
All values in Table 1 satisfy the equation [tex]\( y = 5x \)[/tex].
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } (x) & \text{Money Collected } (y) \\ \hline-20 & -100 \\ \hline 20 & 100 \\ \hline 100 & 500 \\ \hline 109 & 545 \\ \hline \end{array} \][/tex]
- For [tex]\( x = -20 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times -20 = -100 \)[/tex]. (True)
- For [tex]\( x = 20 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 20 = 100 \)[/tex]. (True)
- For [tex]\( x = 100 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 100 = 500 \)[/tex]. (True)
- For [tex]\( x = 109 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 109 = 545 \)[/tex]. (True)
All values in Table 2 satisfy the equation [tex]\( y = 5x \)[/tex].
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } ( x ) & \text{Money Collected } ( y ) \\ \hline 0 & 0 \\ \hline 10 & 50 \\ \hline 51 & 255 \\ \hline 400 & 2000 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 0 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 0 = 0 \)[/tex]. (True)
- For [tex]\( x = 10 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 10 = 50 \)[/tex]. (True)
- For [tex]\( x = 51 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 51 = 255 \)[/tex]. (True)
- For [tex]\( x = 400 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 400 = 2000 \)[/tex]. (True)
All values in Table 3 satisfy the equation [tex]\( y = 5x \)[/tex].
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline \text{Tickets } (x) & \text{Money Collected } (y) \\ \hline 5 & 25 \\ \hline 65 & 350 \\ \hline \end{array} \][/tex]
- For [tex]\( x = 5 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 5 = 25 \)[/tex]. (True)
- For [tex]\( x = 65 \)[/tex], [tex]\( y \)[/tex] should be [tex]\( 5 \times 65 = 325 \)[/tex]. (False)
In Table 4, only the first pair satisfies the equation [tex]\( y = 5x \)[/tex]. The second pair does not.
### Conclusion:
The tables that represent viable solutions for [tex]\( y = 5x \)[/tex] are:
- [1]
- [2]
- [3]
Therefore, Table 1, Table 2, and Table 3 are the viable solutions for the given equation.
