MISSION · Algebra
Writing the Equation of a Line
Three forms of the same line, and when each one saves you work
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LEVEL 2 · LEARN
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The idea in one sentence
A straight line is pinned down by a direction and one point it passes through, and the three standard forms are three ways of writing that same pair of facts.
The three forms
Slope-intercept: \(y = mx + b\). Here \(m\) is the slope and \(b\) is the \(y\)-intercept. Best for graphing and for reading a line off a graph, because both numbers are visible.
Point-slope: \(y - y_1 = m(x - x_1)\). Best for writing a line, because it takes any point \((x_1, y_1)\) and the slope and needs no rearranging first.
Standard: \(Ax + By = C\). Best for systems of equations and for lines that are vertical, which the other two cannot express.
All three describe the same set of points. Converting between them is rearranging and nothing more.
From a point and a slope
Slope 3 through \((2, 5)\). Drop them straight into point-slope:
\[y - 5 = 3(x - 2)\]That is a complete answer. If the question wants slope-intercept, expand and tidy: \(y - 5 = 3x - 6\), so \(y = 3x - 1\).
The temptation is to start from \(y = mx + b\), substitute the point, and solve for \(b\). It works — \(5 = 3(2) + b\) gives \(b = -1\) — but it is an extra equation to solve every time. Point-slope skips it.
From two points
Two points give you the slope, and then you are back to the previous case.
Through \((2, 5)\) and \((6, 13)\):
\[m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2\]Then \(y - 5 = 2(x - 2)\), which tidies to \(y = 2x + 1\).
Use either point — the other one gives \(y - 13 = 2(x - 6)\), which tidies to exactly the same line. That is a free check on your arithmetic, and it is worth doing once.
Parallel and perpendicular
Parallel lines have equal slopes. A line parallel to \(y = 2x + 1\) has slope 2, and only the intercept distinguishes them.
Perpendicular lines have slopes that multiply to \(-1\), so each is the negative reciprocal of the other. Perpendicular to \(y = 2x + 1\) means slope \(-\frac{1}{2}\).
Flip and change the sign — both, not one. The most common error is flipping without the minus, which gives a line at a plausible-looking but wrong angle.
Why \(-1\)? Rotating a right-triangle of rise \(a\) over run \(b\) by a quarter turn swaps the two legs and reverses one of them, turning \(\frac{a}{b}\) into \(-\frac{b}{a}\). Their product is \(-1\).
The two lines that break the pattern
A horizontal line has slope 0: \(y = 4\). Every point has the same height.
A vertical line has no slope at all — the run is zero and you cannot divide by it. It is written \(x = 3\), and it is why standard form exists: \(1x + 0y = 3\) handles it, while \(y = mx + b\) cannot.
These two are perpendicular to each other, and they are the one perpendicular pair whose slopes do not multiply to \(-1\). The rule quietly assumes both slopes exist.
Common mix-ups
- Subtracting the coordinates in different orders. \(\frac{y_2 - y_1}{x_2 - x_1}\) — same point first on both floors, or the sign flips.
- Mixing up run and rise. Slope is rise over run, vertical over horizontal.
- Forgetting the minus in the negative reciprocal.
- Sign errors with negative points. \(y - (-3) = m(x - (-1))\) becomes \(y + 3 = m(x + 1)\). Write the brackets before you simplify them.
- Calling slope 0 “no slope”. Slope 0 is horizontal; undefined slope is vertical.
Check yourself
Write the line through \((0,4)\) with slope 3, find the slope through \((2,5)\) and \((6,13)\), and give the slope of a line perpendicular to \(y = 2x + 1\).
The intercept is given, so \(y = 3x + 4\). The slope is \(\frac{13-5}{6-2} = 2\). And the negative reciprocal of 2 is \(-\frac{1}{2}\).
LEVEL 3 · REFLECT
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