The equation of a line

The three forms

• The same straight line can be written three ways, and you need all three.

• Gradient-intercept form:

$$y = mx + c$$

• $m$ — the gradient

• $c$ — the $y$-intercept, where the line crosses the $y$-axis

• Best for reading off the gradient at a glance, and for sketching.

• Point-gradient form:

$$y - y_1 = m(x - x_1)$$

• $(x_1, y_1)$ — any point on the line

• Best for building an equation from a point and a gradient.

• General form:

$$ax + by + c = 0$$

• All terms on one side, usually with integer coefficients and $a$ positive.
• The only form that can describe a vertical line, which has no gradient.

Building a line from a point and a gradient

• Substitute into the point-gradient form, then rearrange.

• Worked through — the line through $(3, 5)$ with gradient 2:

  • $$y - 5 = 2(x - 3)$$
  • $$y - 5 = 2x - 6$$
  • $$y = 2x - 1$$

Building a line from two points

• Find the gradient first, then use either point in the point-gradient form.

• Worked through — the line through $A(1, 4)$ and $B(5, 12)$:

  • Gradient: $m = \dfrac{12-4}{5-1} = \dfrac{8}{4} = 2$
  • Using $A$: $y - 4 = 2(x - 1)$, so $y = 2x + 2$
  • Using $B$ as a check: $y - 12 = 2(x-5)$, so $y = 2x + 2$ ✓ the same line

• Either point gives the same answer. Using the second one as a check costs nothing.

Horizontal and vertical lines

• Horizontal line through $(a, b)$: every point has the same $y$-value, so the equation is

$$y = b$$

• Vertical line through $(a, b)$: every point has the same $x$-value, so

$$x = a$$

• A vertical line has no $y = mx + c$ form, because its gradient is undefined. It can only be written as $x = a$ (or in general form).
• The commonest confusion: $y = 4$ is horizontal, and $x = 4$ is vertical. Remember that $y = 4$ fixes the height.

Converting between forms

• To general form: multiply out, clear fractions, and move everything to one side.

  • $y = \dfrac{2}{3}x - 1$ → multiply by 3: $3y = 2x - 3$ → $2x - 3y - 3 = 0$

• To gradient-intercept form: make $y$ the subject.

  • $4x + 2y - 8 = 0$ → $2y = -4x + 8$ → $y = -2x + 4$, so $m = -2$ and $c = 4$

• Always convert to $y = mx + c$ before comparing gradients. Two equations in general form can look completely different and still be parallel.

Finding the intercepts

• $y$-intercept: set $x = 0$ and solve for $y$. In $y = mx + c$ it is just $c$.

• $x$-intercept: set $y = 0$ and solve for $x$.

• For $y = 3x - 12$: the $y$-intercept is $-12$, and setting $y = 0$ gives $x = 4$.

Checking a point is on a line

• Substitute the point's coordinates into the equation.
• If both sides are equal, the point lies on the line; if not, it does not.
• For $y = 2x + 1$ and the point $(3, 7)$: $2(3) + 1 = 7$ ✓ the point is on the line.