Fill in the table. Find th rule: $\newline$ $\newline$ $X$ (input) $\newline$ $Y$ (output $\newline$ $\newline$ $1$ $\newline$ $5$ $\newline$ $\newline$ $3$ $\newline$ $9$ $\newline$ $\newline$ $4$ $\newline$ $11$

Full solution

Q. Fill in the table. Find th rule:

\newline

\newline

X

(input)

\newline

Y

(output

\newline

\newline

1

\newline

5

\newline

\newline

3

\newline

9

\newline

\newline

4

\newline

11

Analyze Pattern: Step $1$ : Analyze the given values to find a pattern. $\newline$ We have pairs of values for $X$ and $Y$ : $\newline$ $X = 1$ , $Y = 5$ $\newline$ $X = 3$ , $Y = 9$ $\newline$ $X = 4$ , $Y = 11$ $\newline$ We need to find a rule that connects $X$ to $Y$ . Let's check if a linear relationship fits.
Assume Linear Relationship: Step $2$ : Assume a linear relationship $Y = mX + b$ . $\newline$ Using the first pair $(X=1, Y=5)$ , we get: $\newline$ $5 = m\cdot1 + b$ $\newline$ $\Rightarrow b = 5 - m$ $\newline$ Using the second pair $(X=3, Y=9)$ , we get: $\newline$ $9 = m\cdot3 + (5 - m)$ $\newline$ $\Rightarrow 9 = 2m + 5$ $\newline$ $\Rightarrow 2m = 4$ $\newline$ $\Rightarrow m = 2$
Substitute to Find Rule: Step $3$ : Substitute $m$ back to find $b$ . From $b = 5 - m$ and $m = 2$ : $b = 5 - 2$ $\Rightarrow b = 3$ So, the rule is $Y = 2X + 3$ .
Check Rule with Third Pair: Step $4$ : Check the rule with the third pair $(X=4, Y=11)$ . $\newline$ Substitute $X = 4$ into $Y = 2X + 3$ : $\newline$ $Y = 2\cdot4 + 3$ $\newline$ $\Rightarrow Y = 11$ $\newline$ The rule works for this pair too.
Verify Rule with All Pairs: Step $5$ : Verify the rule with all pairs. $\newline$ For $X = 1$ , $Y = 2 \times 1 + 3 = 5$ $\newline$ For $X = 3$ , $Y = 2 \times 3 + 3 = 9$ $\newline$ For $X = 4$ , $Y = 2 \times 4 + 3 = 11$ $\newline$ All pairs match the rule.