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Sagot :
To determine which table shows the viable solutions for the total number of pages Selma has read after [tex]\( x \)[/tex] minutes, given her pace of reading and initial pages read, we need to confirm that the equation [tex]\( y = 75 + 2x \)[/tex] holds true for all the given data points in each table. Here we start with the initial pages read as 75, and she reads at a rate of 2 pages per minute.
### Analyzing Each Table
Table 1:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline 2 & 79 \\ \hline 14 & 101 \\ \hline 39 & 153 \\ \hline 55 & 185 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = 2, y = 79 \)[/tex]
[tex]\[ y = 75 + 2(2) = 75 + 4 = 79 \quad \text{(Correct)} \][/tex]
2. [tex]\( x = 14, y = 101 \)[/tex]
[tex]\[ y = 75 + 2(14) = 75 + 28 = 103 \quad \text{(Incorrect)} \][/tex]
3. [tex]\( x = 39, y = 153 \)[/tex]
[tex]\[ y = 75 + 2(39) = 75 + 78 = 153 \quad \text{(Correct)} \][/tex]
4. [tex]\( x = 55, y = 185 \)[/tex]
[tex]\[ y = 75 + 2(55) = 75 + 110 = 185 \quad \text{(Correct)} \][/tex]
Since the second entry does not match, Table 1 is not valid.
Table 2:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline -16 & 43 \\ \hline 6 & 87 \\ \hline 27 & 129 \\ \hline 52 & 179 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = -16, y = 43 \)[/tex]
[tex]\[ y = 75 + 2(-16) = 75 - 32 = 43 \quad \text{(Correct)} \][/tex]
2. [tex]\( x = 6, y = 87 \)[/tex]
[tex]\[ y = 75 + 2(6) = 75 + 12 = 87 \quad \text{(Correct)} \][/tex]
3. [tex]\( x = 27, y = 129 \)[/tex]
[tex]\[ y = 75 + 2(27) = 75 + 54 = 129 \quad \text{(Correct)} \][/tex]
4. [tex]\( x = 52, y = 179 \)[/tex]
[tex]\[ y = 75 + 2(52) = 75 + 104 = 179 \quad \text{(Correct)} \][/tex]
All entries match, Table 2 is valid.
Table 3:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline 0 & 0 \\ \hline 19 & 113 \\ \hline 32 & 139 \\ \hline 47 & 169 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = 0, y = 0 \)[/tex]
[tex]\[ y = 75 + 2(0) = 75 + 0 = 75 \quad \text{(Incorrect)} \][/tex]
2. [tex]\( x = 19, y = 113 \)[/tex]
[tex]\[ y = 75 + 2(19) = 75 + 38 = 113 \quad \text{(Correct)} \][/tex]
3. [tex]\( x = 32, y = 139 \)[/tex]
[tex]\[ y = 75 + 2(32) = 75 + 64 = 139 \quad \text{(Correct)} \][/tex]
4. [tex]\( x = 47, y = 169 \)[/tex]
[tex]\[ y = 75 + 2(47) = 75 + 94 = 169 \quad \text{(Correct)} \][/tex]
Since the first entry does not match, Table 3 is not valid.
Table 4:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline 0 & 75 \\ \hline 11 & 97 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = 0, y = 75 \)[/tex]
[tex]\[ y = 75 + 2(0) = 75 + 0 = 75 \quad \text{(Correct)} \][/tex]
2. [tex]\( x = 11, y = 97 \)[/tex]
[tex]\[ y = 75 + 2(11) = 75 + 22 = 97 \quad \text{(Correct)} \][/tex]
All entries match, Table 4 is valid.
### Conclusion
The viable tables that show the total number of pages Selma has read after [tex]\( x \)[/tex] minutes are Table 2 and Table 4.
### Analyzing Each Table
Table 1:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline 2 & 79 \\ \hline 14 & 101 \\ \hline 39 & 153 \\ \hline 55 & 185 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = 2, y = 79 \)[/tex]
[tex]\[ y = 75 + 2(2) = 75 + 4 = 79 \quad \text{(Correct)} \][/tex]
2. [tex]\( x = 14, y = 101 \)[/tex]
[tex]\[ y = 75 + 2(14) = 75 + 28 = 103 \quad \text{(Incorrect)} \][/tex]
3. [tex]\( x = 39, y = 153 \)[/tex]
[tex]\[ y = 75 + 2(39) = 75 + 78 = 153 \quad \text{(Correct)} \][/tex]
4. [tex]\( x = 55, y = 185 \)[/tex]
[tex]\[ y = 75 + 2(55) = 75 + 110 = 185 \quad \text{(Correct)} \][/tex]
Since the second entry does not match, Table 1 is not valid.
Table 2:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline -16 & 43 \\ \hline 6 & 87 \\ \hline 27 & 129 \\ \hline 52 & 179 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = -16, y = 43 \)[/tex]
[tex]\[ y = 75 + 2(-16) = 75 - 32 = 43 \quad \text{(Correct)} \][/tex]
2. [tex]\( x = 6, y = 87 \)[/tex]
[tex]\[ y = 75 + 2(6) = 75 + 12 = 87 \quad \text{(Correct)} \][/tex]
3. [tex]\( x = 27, y = 129 \)[/tex]
[tex]\[ y = 75 + 2(27) = 75 + 54 = 129 \quad \text{(Correct)} \][/tex]
4. [tex]\( x = 52, y = 179 \)[/tex]
[tex]\[ y = 75 + 2(52) = 75 + 104 = 179 \quad \text{(Correct)} \][/tex]
All entries match, Table 2 is valid.
Table 3:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline 0 & 0 \\ \hline 19 & 113 \\ \hline 32 & 139 \\ \hline 47 & 169 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = 0, y = 0 \)[/tex]
[tex]\[ y = 75 + 2(0) = 75 + 0 = 75 \quad \text{(Incorrect)} \][/tex]
2. [tex]\( x = 19, y = 113 \)[/tex]
[tex]\[ y = 75 + 2(19) = 75 + 38 = 113 \quad \text{(Correct)} \][/tex]
3. [tex]\( x = 32, y = 139 \)[/tex]
[tex]\[ y = 75 + 2(32) = 75 + 64 = 139 \quad \text{(Correct)} \][/tex]
4. [tex]\( x = 47, y = 169 \)[/tex]
[tex]\[ y = 75 + 2(47) = 75 + 94 = 169 \quad \text{(Correct)} \][/tex]
Since the first entry does not match, Table 3 is not valid.
Table 4:
[tex]\[ \begin{array}{|c|c|} \hline \text{Minutes of Reading }( x ) & \text{Total Pages Read }( y ) \\ \hline 0 & 75 \\ \hline 11 & 97 \\ \hline \end{array} \][/tex]
For each entry, check if [tex]\( y = 75 + 2x \)[/tex]:
1. [tex]\( x = 0, y = 75 \)[/tex]
[tex]\[ y = 75 + 2(0) = 75 + 0 = 75 \quad \text{(Correct)} \][/tex]
2. [tex]\( x = 11, y = 97 \)[/tex]
[tex]\[ y = 75 + 2(11) = 75 + 22 = 97 \quad \text{(Correct)} \][/tex]
All entries match, Table 4 is valid.
### Conclusion
The viable tables that show the total number of pages Selma has read after [tex]\( x \)[/tex] minutes are Table 2 and Table 4.
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