Skill: Write a rule from a table

A table shows pairs of numbers: an input and its matching output. To write a rule, find the same operation or operations that turn every input into its output.

Example

Input $x$	Output $y$
$1$	$5$
$2$	$8$
$3$	$11$
$4$	$14$

Step 1: Compare the inputs and outputs.

When the input increases by $1$, the output increases by $3$:

• $5$ to $8$: add $3$
• $8$ to $11$: add $3$
• $11$ to $14$: add $3$

This suggests that the input is multiplied by $3$. Now check whether we need another operation.

Step 2: Try multiplying each input by $3$.

• $1 \times 3=3$
• $2 \times 3=6$
• $3 \times 3=9$
• $4 \times 3=12$

Each result is $2$ less than the output, so add $2$.

The rule is

$$y=3x+2$$

In words: Multiply the input by $3$, then add $2$.

Step 3: Verify the rule with every pair.

• If $x=1$, then $3(1)+2=5$
• If $x=2$, then $3(2)+2=8$
• If $x=3$, then $3(3)+2=11$
• If $x=4$, then $3(4)+2=14$

The rule works for every row, so the rule is $\boxed{y=3x+2}$.