Equations of a Line

Try this

Give $P_{1}$ and $P_{2}$ the same $x$ -coordinate (the line stands up straight). Which forms turn into ✗? What is the angle of inclination?
Make the line pass through the origin, then make it parallel to the $x$ -axis. What does the intercept form say in each case?
Keep your eye on the “slope” and the angle of inclination $α$ . When $α$ is acute, is $k$ positive or negative? And when it is obtuse?
However you drag, does the “general” form at the bottom ever show ✗?

Angle of inclination: the smallest positive angle from the positive direction of the $x$ -axis to the upward direction of the line, $0^{\circ} \leq α < 18 0^{\circ}$ (a horizontal line has $α = 0^{\circ}$ by convention). When $α \neq = 9 0^{\circ}$ , its tangent is called the slope:

k = tan α = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} (x_{1} \neq = x_{2}) .

This gives five ways to write a line: the point–slope form $y - y_{1} = k (x - x_{1})$ , the slope–intercept form $y = k x + b$ , the two-point form, the intercept form $\frac{x}{a} + \frac{y}{b} = 1$ , and the general form $A x + B y + C = 0$ . The first four each work only for some lines; the general form has no such limit:

The equation of any line in the plane is a linear equation in $x$ and $y$ ; conversely, the graph of any linear equation $A x + B y + C = 0$ (with $A$ , $B$ not both $0$ ) is a line.

Predict first

What is the area of the triangle enclosed by the line $3 x - 4 y + 12 = 0$ and the two coordinate axes?

Choose an answer first; then you can see the result.

Why: one line, five ways to write it

Example: the line through (−1, −3) and (2, 3)

Step 1 / 5

Find the slope first
From $P_{1}$ to $P_{2}$ : $3$ to the right and $6$ up, so $k = \frac{6}{3} = 2$ . The angle of inclination $α$ satisfies $tan α = 2$ , so it is acute.

$k = \frac{3 - ( - 3 )}{2 - ( - 1 )} = 2$
Point–slope: one point + the slope
Slope–intercept: solve for y
Intercept form: the two intercepts
General form: move everything to one side

Why does the general form “miss none”? The proof needs just two cases:

Proof: Ax + By + C = 0 (A, B not both 0) always describes a line

Step 1 / 3

Case 1: B ≠ 0
Divide both sides by $B$ and solve for $y$ . This is the slope–intercept form with slope $- \frac{A}{B}$ and intercept $- \frac{C}{B}$ : a line not perpendicular to the $x$ -axis.

$y = - \frac{A}{B} x - \frac{C}{B}$
Case 2: B = 0
Conversely

$A x + B y + C = 0$ has three coefficients, but multiplying them all by the same nonzero number gives the same line, so only two ratios are really free. That is why a line is determined by two independent conditions: two points, or a point and a slope, or two intercepts…

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The line can be written in intercept form $\frac{x}{a} + \frac{y}{b} = 1$

When the conditions hold, both intercepts exist and are nonzero, so the intercept form can always be written.

Practice

Exercise 1Basic

Find the angle of inclination (in degrees) of the line through $(1, - 2)$ and $(4, 1)$ .

Exercise 2Basic

Find the general-form equation of the line through $(3, 1)$ with slope $- 2$ (positive coefficient of $x$ , coefficients with no common factor).

Exercise 3Advanced

A line passes through $(- 2, 3)$ , and its two intercepts add up to $2$ . Find all possible values of its $x$ -intercept $a$ (separate them with commas).

Exercise 4Challenge

A line $l$ through $(1, 2)$ meets the positive $x$ -axis at $A$ and the positive $y$ -axis at $B$ . Find the minimum area of $△ A O B$ .

Try this

Why: one line, five ways to write it

Example: the line through (−1, −3) and (2, 3)

Proof: Ax + By + C = 0 (A, B not both 0) always describes a line

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

“Equal intercepts”, but only the intercept form was tried

Saying a vertical line has slope 0