Write equations of lines in slope-intercept form

Slope-intercept form, $y=mx+b$ , expresses a nonvertical linear relationship in which $m$ is the constant rate of change or slope and $b$ is the $y$ -intercept, allowing an equation to be constructed from a graph, a table, a slope and point, or two points. The learner connects algebraic, graphical, and contextual representations, understands that equivalent equations describe the same line, and distinguishes the $y$ -intercept from an arbitrary point on the line; vertical lines and more advanced forms or generalizations are not included.

Detailed Explanation: Write equations of lines in slope-intercept form

A line in slope-intercept form is written as

y=mx+b

where:

$m$ is the slope, or rate of change
$b$ is the $y$ -intercept, the point where the line crosses the $y$ -axis

Example

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

Step 1: Find the slope.

Use

m=\frac{y_2-y_1}{x_2-x_1}

Substitute the two points:

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

So the equation begins as

y=2x+b

Step 2: Find $b$ using one point.

Use $(2,5)$ , so $x=2$ and $y=5$ :

5=2(2)+b

Solve for $b$ :

5=4+b

b=1

Step 3: Write the equation.

Substitute $m=2$ and $b=1$ into $y=mx+b$ :

\boxed{y=2x+1}

Step 4: Check the other point.

For $(6,13)$ :

y=2(6)+1=13

The equation works for both points. Notice that the $y$ -intercept is $(0,1)$ , not either of the given points.

Learn by doing: Write equations of lines in slope-intercept form

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Equation of a Line from Coordinates of Points

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