Write an equation from a table

A table of corresponding $x$ - and $y$ -values represents a linear relation when equal changes in $x$ produce a constant change in $y$ ; this constant rate of change is the slope. The relation can be expressed as $y=mx+b$ , with $m$ determined from the table and $b$ representing the value of $y$ when $x=0$ , whether or not the line passes through the origin. This scope excludes nonlinear, piecewise, and more abstract parameterized relations.

Detailed Explanation: 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.

Learn by doing: Write an equation from a table

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Algebra - Find Equivalent - X,Y Chart to Function

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