Equations from Two Points: Write, Derive, and Check

Call the two different points $(x_1, y_1)$ and $(x_2, y_2)$. Going from the first point to the second, $y$ changes by $y_2 - y_1$ and $x$ changes by $x_2 - x_1$. When the two $x$-values are different, the slope is the change in $y$ divided by the change in $x$. Written as a fraction, the subtractions must go in the same order on the top and on the bottom. Either point can come first, as long as it comes first in both places.

When the two $x$-values are equal, the line is vertical (straight up and down). The change in $x$ is $0$, and you cannot divide by $0$, so the slope of a vertical line is undefined. A vertical line also pairs one $x$-value with many $y$-values, so it is not a function $y = f(x)$, which is a rule that gives exactly one $y$ for each $x$. It has no equation of the form $y = mx + b$, so the rest of this lesson is about lines that are not vertical.

An equation of a line must be true for the coordinates of every point on that line. That tells you how to finish once you know $m$. Only $b$ is missing from $y = mx + b$, so replace $x$ and $y$ with the coordinates of one of your points (this is called substituting) and solve for $b$. Then test your equation with the other point. If both points make $y = mx + b$ true, its graph is a line through both of them. Only one line passes through two different points, so your equation is right.

Why does one equation fit every point on a line, not just the two you were given? The reason is that on a line that is not vertical, the slope between any two different points is the same number. You have relied on this whenever you read a slope from a graph.

Take a line through the origin, the point $(0, 0)$, with slope $m$. Pick any other point $(x, y)$ on it. A line that is not vertical crosses the $y$-axis only once, so this point has $x \neq 0$. The slope from $(0, 0)$ to $(x, y)$ is $\frac{y - 0}{x - 0} = \frac{y}{x}$, and it must equal $m$. Multiply both sides of $\frac{y}{x} = m$ by $x$, which keeps the equation true, to get $y = mx$. The origin fits too, because $0 = m \cdot 0$. So every point on the line makes $y = mx$ true.

In the figure, the horizontal axis shows $x$ and the vertical axis shows $y$. Line B passes through $(0, 0)$ with slope $2$: its points $(2, 4)$ and $(4, 8)$ give $\frac{4}{2} = 2$ and $\frac{8}{4} = 2$, so Line B is $y = 2x$.

An equation of a line must be true for every point on that line, so it must be true for both points you were given. That gives you a reliable way to find a mistake: substitute each given point into the finished equation. If either point makes it false, something went wrong. Then ask:

• Were the two subtractions for the slope done in the same order?
• Was subtracting a negative number treated as adding its opposite?
• Does the sign of the slope match the points? If $y$ goes down as $x$ goes up, the slope must be negative.
• Was $b$ solved for, or just copied from a point whose $x$-value is not $0$?

Be careful with a one-point test. If $b$ was solved correctly from one point, that point fits the equation even when the slope is wrong, so always test the other point too. Mateo's work in the table has a mistake in it. Try to find it before you read the example after the table.

The big idea: a line that is not vertical has one constant slope, and everything in this lesson grows from that fact. Two points with different $x$-values give you $m$, and substituting one of them into $y = mx + b$ gives you $b$. The same constant slope is why every point on the line fits $y = mx + b$, or $y = mx$ when the line passes through the origin. To catch a mistake, test both given points in the finished equation: if one fails, recheck the slope first and then $b$.