Write the equation of a line in slope-intercept and point-slope form from a slope and point, two points, or a graph (Ohio A-CED.2, F-IF, F-LE).

What this topic is asking

Ohio standard A-CED.2 asks you to write the equation of a line in two variables from the information you are given: a slope and a point, two points, or a graph. The two forms you need, slope-intercept and point-slope, are both on the reference sheet. This skill sits across the Expressions and Equations and Functions reporting categories and is heavily tested.

The two line forms

Both forms describe the same lines; you pick based on what you are given.

If you are handed the slope and the $y$-intercept, slope-intercept is immediate. If you are handed the slope and a point that is not on the $y$-axis, point-slope is cleaner, and you can rearrange to slope-intercept afterward.

From two points

When two points are given, the slope comes first.

It does not matter which of the two points you use in point-slope; both give the same final equation.

Parallel and perpendicular lines

The slope encodes direction, so it controls these relationships.

• Parallel lines never meet and have the same slope. A line parallel to $y = 2x + 1$ has slope $2$.
• Perpendicular lines meet at a right angle and have slopes that are negative reciprocals: if one slope is $m$, the other is $-\dfrac{1}{m}$. A line perpendicular to $y = 2x + 1$ has slope $-\dfrac{1}{2}$.

To write such a line, take the correct slope and then use a given point with point-slope form.

How Ohio examines this topic

• Equation response. Type the equation in the requested form (slope-intercept or point-slope).
• Multiple choice. Pick the correct equation from a graph or from given conditions.
• Multi-part items. Find a slope, then build the equation, then use it to predict a value.

The reference sheet supplies both forms, so the credit is for choosing and applying the right one.

Why point-slope is the all-purpose tool

Slope-intercept gets the spotlight, but point-slope is the more flexible starting point because it works from any point, not just the $y$-intercept. Many problems give you a slope and a point like $(2, 1)$ that is not on the $y$-axis; forcing slope-intercept here means solving for $b$ first, an extra step, while point-slope lets you write the line immediately and tidy up afterward. The form simply encodes the definition of slope: $\frac{y - y_1}{x - x_1} = m$ rearranged. Once you are comfortable converting point-slope to slope-intercept by distributing and isolating $y$, you can handle every "write the line" prompt with one reliable method and only switch to slope-intercept when the $y$-intercept is handed to you directly.

Reading a line off a graph

When the line is given as a graph, recover the two pieces it needs. The $y$-intercept is where the line crosses the $y$-axis, giving $b$ directly. The slope is rise over run between two clear lattice points: count the vertical change and divide by the horizontal change, keeping the sign (down-to-the-right is negative). With $b$ and $m$ in hand, write $y = mx + b$. This graph-reading skill connects writing equations to interpreting key features of graphs, and it is exactly what a "match the graph to its equation" item checks.

Try this

Q1. Write the line with slope $\frac{1}{2}$ through $(0, -3)$ in slope-intercept form. [2 points]

• Cue. $b = -3$, so $y = \frac{1}{2}x - 3$.

Q2. Write the line through $(-1, 4)$ and $(2, -2)$ in slope-intercept form. [2 points]

• Cue. $m = \frac{-2 - 4}{2 - (-1)} = \frac{-6}{3} = -2$; $y - 4 = -2(x + 1)$, so $y = -2x + 2$.