Answer :
To determine which table of values follows the rule "five less than triple a number," we need to verify the given relation:
[tex]\[ y = 3x - 5 \][/tex]
We will check each table step-by-step to see if the relation holds true for all given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] pairs.
Table 1:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & -11 & -8 & -5 & -2 & 1 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 3(-2) - 5 = -6 - 5 = -11 \][/tex]
Correct, [tex]\( y = -11 \)[/tex].
- If [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 3(-1) - 5 = -3 - 5 = -8 \][/tex]
Correct, [tex]\( y = -8 \)[/tex].
- If [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0) - 5 = 0 - 5 = -5 \][/tex]
Correct, [tex]\( y = -5 \)[/tex].
- If [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1) - 5 = 3 - 5 = -2 \][/tex]
Correct, [tex]\( y = -2 \)[/tex].
- If [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 3(2) - 5 = 6 - 5 = 1 \][/tex]
Correct, [tex]\( y = 1 \)[/tex].
All values in Table 1 satisfy the rule [tex]\( y = 3x - 5 \)[/tex].
Table 2:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -17 & -11 & -5 & 1 & 7 \\ \hline y & -4 & -2 & 0 & 2 & 4 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -17 \)[/tex]:
[tex]\[ y = 3(-17) - 5 = -51 - 5 = -56 \][/tex]
Incorrect, [tex]\( y \neq -4 \)[/tex].
Since one of the values does not match, Table 2 does not satisfy the rule [tex]\( y = 3x - 5 \)[/tex]. There's no need to check the rest.
Table 3:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -5 & -3 & -1 & 1 & 3 \\ \hline y & -20 & -14 & -8 & 2 & 12 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -5 \)[/tex]:
[tex]\[ y = 3(-5) - 5 = -15 - 5 = -20 \][/tex]
Correct, [tex]\( y = -20 \)[/tex].
- If [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 3(-3) - 5 = -9 - 5 = -14 \][/tex]
Correct, [tex]\( y = -14 \)[/tex].
- If [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 3(-1) - 5 = -3 - 5 = -8 \][/tex]
Correct, [tex]\( y = -8 \)[/tex].
- If [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1) - 5 = 3 - 5 = -2 \][/tex]
Incorrect, [tex]\( y \neq 2 \)[/tex].
Since one of the values does not match, Table 3 does not satisfy the rule [tex]\( y = 3x - 5 \)[/tex].
Table 4:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 \\ \hline y & -8 & -3 & 2 & 7 & 12 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 3(-1) - 5 = -3 - 5 = -8 \][/tex]
Correct, [tex]\( y = -8 \)[/tex].
- If [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0) - 5 = 0 - 5 = -5 \][/tex]
Incorrect, [tex]\( y \neq -3 \)[/tex].
Since one of the values does not match, Table 4 does not satisfy the rule [tex]\( y = 3x - 5 \)[/tex].
From our verification, the table that follows the rule [tex]\( y = 3x - 5 \)[/tex] is Table 1.
[tex]\[ y = 3x - 5 \][/tex]
We will check each table step-by-step to see if the relation holds true for all given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] pairs.
Table 1:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & -11 & -8 & -5 & -2 & 1 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -2 \)[/tex]:
[tex]\[ y = 3(-2) - 5 = -6 - 5 = -11 \][/tex]
Correct, [tex]\( y = -11 \)[/tex].
- If [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 3(-1) - 5 = -3 - 5 = -8 \][/tex]
Correct, [tex]\( y = -8 \)[/tex].
- If [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0) - 5 = 0 - 5 = -5 \][/tex]
Correct, [tex]\( y = -5 \)[/tex].
- If [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1) - 5 = 3 - 5 = -2 \][/tex]
Correct, [tex]\( y = -2 \)[/tex].
- If [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 3(2) - 5 = 6 - 5 = 1 \][/tex]
Correct, [tex]\( y = 1 \)[/tex].
All values in Table 1 satisfy the rule [tex]\( y = 3x - 5 \)[/tex].
Table 2:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -17 & -11 & -5 & 1 & 7 \\ \hline y & -4 & -2 & 0 & 2 & 4 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -17 \)[/tex]:
[tex]\[ y = 3(-17) - 5 = -51 - 5 = -56 \][/tex]
Incorrect, [tex]\( y \neq -4 \)[/tex].
Since one of the values does not match, Table 2 does not satisfy the rule [tex]\( y = 3x - 5 \)[/tex]. There's no need to check the rest.
Table 3:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -5 & -3 & -1 & 1 & 3 \\ \hline y & -20 & -14 & -8 & 2 & 12 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -5 \)[/tex]:
[tex]\[ y = 3(-5) - 5 = -15 - 5 = -20 \][/tex]
Correct, [tex]\( y = -20 \)[/tex].
- If [tex]\( x = -3 \)[/tex]:
[tex]\[ y = 3(-3) - 5 = -9 - 5 = -14 \][/tex]
Correct, [tex]\( y = -14 \)[/tex].
- If [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 3(-1) - 5 = -3 - 5 = -8 \][/tex]
Correct, [tex]\( y = -8 \)[/tex].
- If [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(1) - 5 = 3 - 5 = -2 \][/tex]
Incorrect, [tex]\( y \neq 2 \)[/tex].
Since one of the values does not match, Table 3 does not satisfy the rule [tex]\( y = 3x - 5 \)[/tex].
Table 4:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 \\ \hline y & -8 & -3 & 2 & 7 & 12 \\ \hline \end{array} \][/tex]
For each [tex]\( x \)[/tex]:
- If [tex]\( x = -1 \)[/tex]:
[tex]\[ y = 3(-1) - 5 = -3 - 5 = -8 \][/tex]
Correct, [tex]\( y = -8 \)[/tex].
- If [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 3(0) - 5 = 0 - 5 = -5 \][/tex]
Incorrect, [tex]\( y \neq -3 \)[/tex].
Since one of the values does not match, Table 4 does not satisfy the rule [tex]\( y = 3x - 5 \)[/tex].
From our verification, the table that follows the rule [tex]\( y = 3x - 5 \)[/tex] is Table 1.
