Writing Equations of Lines

The y-intercept of a line is the $y$-coordinate of the point where the line crosses the $y$-axis, the point with $x = 0$. Every nonvertical line crosses the $y$-axis exactly once.

Given: a nonvertical line has slope $m$ and $y$-intercept $b$, so $(0, b)$ is on it. Take any other point $(x, y)$ on the line. Because the line is nonvertical, no two of its points share an $x$-coordinate, so $x \ne 0$. The definition of slope gives $\frac{y - b}{x - 0} = m$, which is $\frac{y - b}{x} = m$. Multiply both sides by the nonzero number $x$ to get $y - b = mx$. Then add $b$ to both sides to get slope-intercept form, shown below. The point $(0, b)$ satisfies it too, because $m(0) + b = b$.

So every point on the line satisfies the equation. The steps also run backward, so the converse, the statement with its two parts swapped, holds too: a point that satisfies the equation is on the line.

When you know the slope and one point, substitute them into this form and solve for $b$.

In the figure above, the horizontal axis is the $x$-axis and the vertical axis is the $y$-axis. Line A is the line $y = 2x - 5$. Point A is the given point $(3, 1)$. Point B is $(0, -5)$, because $x = 0$ gives $y = 2(0) - 5 = -5$. So $b = -5$ is the $y$-coordinate at $x = 0$, while the given point's $y$-coordinate is $1$.

The slope matches too. From Point B to Point A, the change in $y$ is $1 - (-5) = 6$ and the change in $x$ is $3 - 0 = 3$, so the slope is $\frac{6}{3} = 2$. These facts come from the stated coordinates and the algebra, not from how the drawing looks.

Now suppose you know two points instead of a slope. By the postulate that through any two different points there is exactly one line, those points fix one line. A postulate is a statement accepted without proof. If the points $(x_1, y_1)$ and $(x_2, y_2)$ have $x_1 \ne x_2$, the change in $x$ is nonzero, so the definition of slope gives $m = \frac{y_2 - y_1}{x_2 - x_1}$. You then have a slope and a point, the same problem you just solved. Either given point works, because both are on the line.

Two different points can also give a slope of $0$ or no slope at all, so compare their coordinates before you start. In the figure below, the horizontal axis is the $x$-axis and the vertical axis is the $y$-axis.

Point A at $(-1, 4)$ and Point B at $(3, 4)$ share a $y$-coordinate. Their slope is $\frac{4 - 4}{3 - (-1)} = \frac{0}{4} = 0$, so Line A through them is horizontal. Substituting $(3, 4)$ into $y = 0x + b$ gives $4 = 0(3) + b$, so $b = 4$ and the equation is $y = 4$.

Point C at $(2, -1)$ and Point D at $(2, 3)$ share an $x$-coordinate. The change in $x$ is $2 - 2 = 0$, so the slope is undefined and Line B through them is vertical. Slope-intercept form needs a slope, so it does not apply. Every point on Line B has $x$-coordinate $2$, and every point with $x$-coordinate $2$ is on Line B, so by the definition of an equation of a line, its equation is $x = 2$.

Point-slope form writes a line's equation directly from its slope and one of its points, so you do not need to find $b$.

Given: a nonvertical line has slope $m$ and passes through $(x_1, y_1)$. To show: a point $(x, y)$ is on the line exactly when it satisfies the equation displayed below.

First, take any other point $(x, y)$ on the line. Because the line is nonvertical, no two of its points share an $x$-coordinate, so $x \ne x_1$. The definition of slope gives $\frac{y - y_1}{x - x_1} = m$. Multiplying both sides by the nonzero number $x - x_1$ gives $y - y_1 = m(x - x_1)$. The point $(x_1, y_1)$ satisfies this too, because both sides equal $0$.

For the converse, start with a point that satisfies the equation. If $x \ne x_1$, dividing both sides by $x - x_1$ shows that its slope from $(x_1, y_1)$ is $m$, so the key concept puts it on the line. If $x = x_1$, the equation forces $y = y_1$, so it is the known point. A vertical line has no slope to use as $m$, so this form does not apply to it.

Takeaway: a slope and a point fix a nonvertical line, and two points with different $x$-coordinates give you that slope. You can write the line as $y = mx + b$ by solving for $b$, or as $y - y_1 = m(x - x_1)$ by substituting directly. If two different points share an $x$-coordinate, the slope is undefined, the line is vertical, and its equation is $x = c$, where $c$ is that shared coordinate. Next, you will write the equation of a line parallel or perpendicular to a given line through a point.