Skill: Write an equation from a table

A table represents a linear relation if equal changes in $x$ produce equal changes in $y$. To write its equation, use

$$y=mx+b,$$

where:

• $m$ is the slope, or rate of change
• $b$ is the value of $y$ when $x=0$

Example

Write an equation for the relation in the table.

$x$	$y$
1	4
3	10
5	16

Step 1: Find the changes in $x$ and $y$.

From $x=1$ to $x=3$, the change in $x$ is $2$.

From $y=4$ to $y=10$, the change in $y$ is $6$.

Check the next pair:

• $x$ changes by $2$
• $y$ changes by $6$

The changes are constant, so the relation is linear.

Step 2: Find the slope.

The slope is

$$m=\frac{\text{change in }y}{\text{change in }x} =\frac{6}{2}=3.$$

So far, the equation is

$$y=3x+b.$$

Step 3: Find $b$.

Use any pair from the table, such as $(1,4)$. Substitute $x=1$ and $y=4$:

$$4=3(1)+b$$ $$4=3+b$$ $$b=1.$$

Step 4: Write the equation.

Substitute $m=3$ and $b=1$ into $y=mx+b$:

$$\boxed{y=3x+1}$$

You can check the equation using another table value. When $x=5$,

$$y=3(5)+1=16,$$

which matches the table.