Find the gradient of the line 2y - 6x = 5.

Step-by-step solution

1. 1

   Know where the gradient is visible. In the form $y=mx+c$ the gradient is literally the number multiplying $x$. In the given form $2y-6x=5$ it is hidden, so the whole job is to rearrange — no gradient formula with two points is needed.

2. 2

   Move the $x$-term to the right-hand side. Adding $6x$ to both sides:

   $2y=6x+5$

   The $y$-term is now alone, but it still carries a coefficient of $2$.

3. 3

   Divide every term by the coefficient of $y$. Dividing by $2$:

   $y=3x+\frac{5}{2}$

   Dividing the constant as well is the step most often forgotten; skipping it changes $c$ but not $m$, so it does not affect this answer — it would matter if the intercept were asked for.

4. 4

   Read off the gradient and verify it with two points on the line. The coefficient of $x$ gives $m=3$. Checking directly from the original equation: at $x=0$, $2y=5$ so $y=2.5$; at $x=1$, $2y=11$ so $y=5.5$. The gradient between $(0,2.5)$ and $(1,5.5)$ is

   $m=\frac{5.5-2.5}{1-0}=3$

   which matches, so the rearrangement was done correctly.