Linear Function Solver

A linear function changes by the same amount for every equal step in $x$. That constant rate of change is the slope, and it is what makes the graph a straight line.

The algebraic test is about the shape of the equation. Every variable must appear to the first power, alone, and never in a denominator or under a root:

The numerical test is just as reliable: if equally spaced $x$ values give $y$ values with a constant first difference, the relationship is linear.

$\text{slope-intercept: } y = mx + b \qquad \text{point-slope: } y - y_1 = m(x - x_1) \qquad \text{standard: } Ax + By = C$

From two points, $(x_1, y_1)$ and $(x_2, y_2)$:

1. Slope $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ — rise over run.
2. Put $m$ and either point into point-slope form.
3. Expand and rearrange into $y = mx + b$.

From a table, the slope is the change in $y$ divided by the change in $x$ between consecutive rows, and $b$ is the $y$ value at $x = 0$.

To graph $y = mx + b$, plot $(0, b)$, then step 1 right and $m$ up (down if $m$ is negative) to get a second point, and draw the line through them. Slope written as a fraction is a ready-made instruction: $m = -\tfrac{2}{3}$ means down 2, right 3.

• Inverting the slope. It is $\frac{\Delta y}{\Delta x}$, rise over run. Using run over rise gives the reciprocal, a different line.
• Subtracting the coordinates in different orders. If the numerator is $y_2 - y_1$, the denominator must be $x_2 - x_1$ — the same point first in both.
• Assuming $b$ is the last number you see. In $2x + 3y = 6$ the intercept is $y = 2$, not 6; you have to solve for $y$ first.
• Confusing $y = 5$ with $x = 5$. The first is a horizontal line and a perfectly good linear function; the second is vertical, has undefined slope, and is not a function at all.
• Calling a curve linear because it rises steadily. $y = x^2$ increases the whole way for $x > 0$, but its first differences grow, so it is not linear.