Point slope Equation for Straight Lines - Derived in Four Cases

In this lesson, we explain how equations for lines in the coordinate plane follow from similarity of triangles in the plane.

We will see or show below that points (x,y) on non-vertical line L satisfy point-slope equation

y - y₁ = m(x-x₁)

where m the slope is a proportionality constant, a real number, and (x₁, y₁) is a given point on the line. The equation of vertical line through a point (x₁, y₁) is

x = x₁

The derivation of the above two equations for four cases

1. Downward Slanted Line: y - y₁ = m(x-x₁)

2. Upward Slanted Line: y - y₁ = m(x-x₁)

3. Horizontal Line: y - y₁ = 0

4. Vertical Line: x - x₁= 0

follow below.

Downward Slanted Lines - Case 1 of 4

For a downward slanting line, every pair of triangles with hypotenuse on the line and sides parallel to the coordinate axes are similar. Similarity follows because the line through the hypotenuses serves as a transversal for the parallel lines through the horizontal and vertical sides. From latter corresponding angles are equal.

Two triangles one with sides lengths given by lower case and another with lengths in uppercase characters.

---

Similarity implies corresponding sides are proportional. That is for some constant q

drop = q DROP and run = q RUN

The latter gives

Hence the drop over run ratios for the triangles are equal.

The common value of these ratios provides a proportionality constant k such that the drop in each triangle is proportional to the run of each triangle.

Calculation of the drop over run proportionality constant.

The proportionality constant k can be computed from any pair of points (x₁, y₁) and (x₂, y₂) are the line.

From the diagram, the drop over rise proportionality constant is

k

=

DROP RUN

=

That is drop over rise proportionality constant k is given by the calculation

k

=

Equation that follows from proportionality

Now if (x, y) is on the line with drop over run proportionality constant k and (x₁, y₁) is a point on the with x₁ < x on the line we have a situation like the following.

Diagram for x₁ < x

For the run and drop triangle determine by the two points (x,y) and (x₁, y₁), the proportionality constant

k

=

Therefore (x, y) satisfies the equation

y₁ - y = k(x-x₁)

or equivalently

y - y₁ = -k(x-x₁)

where (x₁, y₁) and k are unknown. The change of notation m = -k puts this equation in the sought after form

y - y₁ = m(x-x₁)

The value of m is called the slope of the line. Here m = -k is negative as the drop over run ratio or proportionality constant k is positive - a situation implied by non-zero drop. Positive slopes will be seen below in the case of upward slanting lines.

Exercise: Show the same equation follows if (x, y) and (x₁, y₁) are two points on a line with proportionality constant k = -m with x > x₁

Diagram for x₁ > x

Lines Slanted Upward - Case 2 of 4

For a upward slanting line, every pair of triangles with hypotenuse on the line and sides parallel to the coordinate axes are similar. Similarity follows because the line through the hypotenuses serves as a transversal for the parallel lines through the horizontal and vertical sides. From latter corresponding angles are equal.

Similarity implies the rise over run ratios for the triangles are equal.

The common value of these ratios provides a proportionality constant K such that the rise in each triangle is proportional to the run of each triangle.

Calculation of the Proportionality Constant

The proportionality constant K can be computed from any pair of points (x₁, y₁) and (x₂, y₂) are the line.

Therefore

Equation that follows from proportionality

The assumption that (x,y) is on the line in place of (x ,y)

leads to the equation

where the proportionality constant K is known. The latter equation in turn implies

y - y₁ = K(x-x₁)

The latter gives an equation satisfied by all points on the line through point (x₁, y₁) with rise over run proportionality constant K. The point-slope equation for a line

y - y₁ = m(x-x₁)

appears if we let m = K.

Horizontal Line - Case 3 of 4

Vertical Line - Case 4 of 4.

Conclusion

For each non-vertical slanted line through a point (x₁, y₁), there is a constant m called the slope of the line such that every point (x,y) on the line satisfies the equation

y - y₁ = m(x-x₁)

Note if we call x-coordinate difference (x-x₁) a runand vertical line difference y - y₁ a rise, differences that may be positive, zero or negative, the point slope equation say the rise is proportional to the run with proportionality constant m.

Remark: If we know points (x₁, y₁) and (x₂, y₂) belong to non-vertical line with equation

y - y₁ = m(x-x₁)

then (x,y) = (x₂, y₂) must satisfy the equation. So

y₂ - y₁ = m(x₂-x₁)

Thus the value of m, known or not, must be given by the equation

where the RISE = y₂ - y₁ and RUN = x₂-x₁ are allowed to be real numbers. The slope m provides the (signed) rise over (signed) run proportionality constant or ratio for the line.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science