Common mistakes

Why do I get the wrong slope?

These are common mistakes with lines like $y = 2x + 1$ . A quick check for any point: put its $x$ into the equation and see whether you get its $y$ .

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Mistakes, and what to do instead

Mistake 1 · Why do I get a slope of $\frac{1}{2}$ between $(1, 3)$ and $(4, 9)$ ?
Why it happens
The run was divided by the rise: $\frac{4 - 1}{9 - 3} = \frac{3}{6}$ .
What to do instead
Slope is rise over run: $\frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$ .
Mistake 2 · Why do I get a slope of $-2$ ?
Why it happens
You subtracted the coordinates in opposite orders on top and bottom: $\frac{9 - 3}{1 - 4}$ .
What to do instead
Keep the same order on top and bottom: $\frac{9 - 3}{4 - 1} = 2$ .
Mistake 3 · Why do I think $y = 2x + 1$ crosses the $y$ -axis at $2$ ?
Why it happens
The slope and the intercept got swapped.
What to do instead
In $y = mx + b$ , $b$ is the intercept: the line crosses at $(0, 1)$ .
Mistake 4 · Why do I think $(2, 4)$ is on $y = 2x + 1$ ?
Why it happens
The $+1$ got dropped: $2 \cdot 2 = 4$ is only part of it.
What to do instead
Put in $x = 2$ : $2 \cdot 2 + 1 = 5$ , not $4$ — so it isn’t on the line.

A check you can always do alone

For any point $(x, y)$ , put $x$ into the equation. If you get $y$ , the point is on the line: for $y = 2x + 1$ , $x = 4$ gives $9$ , so $(4, 9)$ is on it.

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