Write a linear equation in slope-intercept form

A nonvertical linear relationship can be represented as $y=mx+b$ , where $m$ is the constant rate of change (slope) and $b$ is the $y$ -intercept, the output when $x=0$ . The equation may be constructed from a graph, table, or two points by relating changes in the variables and identifying the intercept, while preserving the connection among symbolic, graphical, and contextual representations; vertical lines and nonlinear relationships are not included because they cannot be expressed in this form.

Detailed Explanation: Write a linear equation in slope-intercept form

A linear equation in slope-intercept form is

y=mx+b

where:

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

Example

Write the equation of the line passing through $(2,5)$ and $(6,13)$ .

Step 1: Find the slope.

Use

m=\frac{\text{change in }y}{\text{change in }x} =\frac{y_2-y_1}{x_2-x_1}

Substitute the two points:

m=\frac{13-5}{6-2}=\frac{8}{4}=2

The slope is $2$ . This means $y$ increases by $2$ for every increase of $1$ in $x$ .

Step 2: Start with slope-intercept form.

y=mx+b

Substitute $m=2$ :

y=2x+b

Step 3: Find the $y$ -intercept.

Use either given point. Substitute $(2,5)$ :

5=2(2)+b

5=4+b

b=1

So the line crosses the $y$ -axis at $(0,1)$ .

Step 4: Write the equation.

\boxed{y=2x+1}

Check: If $x=6$ , then

y=2(6)+1=13

This matches the second point, so the equation is correct.

Learn by doing: Write a linear equation in slope-intercept form

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Equation of a Line from Coordinates of Points

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