Q. how to find y-intercept with two points

How to find the y-intercept from two points (step-by-step)

Given two distinct points on a line, (x₁, y₁) and (x₂, y₂), the y-intercept is the value b in the line equation \(y = m x + b\). The following steps show exactly what to do and why at each step.

1. Step 1 — Check for a vertical line (special case)

   What to do: Compare x₁ and x₂.

   Why: If x₁ = x₂, the line is vertical and has the form \(x = c\) with \(c = x_1\). A vertical line has no finite y-intercept unless \(c = 0\). If \(c = 0\) then the line is the y-axis and every point on it intersects the y-axis (the concept of a single y-intercept is not meaningful).

   Action:

   If x₁ = x₂ and x₁ ≠ 0, stop: there is no y-intercept. If x₁ = x₂ = 0, the line is the y-axis (every y is an intersection). Otherwise continue to Step 2.

2. Step 2 — Compute the slope m

   What to do: Use the slope formula

   \(m = \dfrac{y_{2} – y_{1}}{x_{2} – x_{1}}\)

   Why: The slope gives the rate of change of y with respect to x and is needed to write the line in slope-intercept form \(y = m x + b\).

   Note: This is valid because x₁ ≠ x₂ (we excluded the vertical-line case in Step 1).

3. Step 3 — Write the line equation and solve for b

   What to do: Start with the slope-intercept form

   \(y = m x + b\)

   Substitute one of the given points, for example (x₁, y₁), and solve for b:

   \(b = y_{1} – m x_{1}\)

   Why: Substituting a known point into \(y = m x + b\) isolates b so you can compute the y-intercept directly.

4. Step 4 — (Optional) Use a single formula for b in terms of the two points

   What to do: Combine the slope formula and the expression for b to get a direct formula for b:

   \(b = y_{1} – x_{1}\,\dfrac{y_{2} – y_{1}}{x_{2} – x_{1}}\)

   Simplify the expression to a single fraction if desired:

   \(b = \dfrac{x_{2}y_{1} – x_{1}y_{2}}{x_{2} – x_{1}}\)

   Why: This formula computes the y-intercept directly from the coordinates of the two points without separately computing m first.

5. Step 5 — State the y-intercept

   What to do: The y-intercept is the point where x = 0, so it is (0, b). The numeric value of the intercept on the y-axis is b as computed above.

   Why: By definition the y-intercept is the y-value when x = 0, which is exactly b in \(y = m x + b\).

Compact summary

For non-vertical lines through (x₁, y₁) and (x₂, y₂):

\(m = \dfrac{y_{2} – y_{1}}{x_{2} – x_{1}}\)

\(b = y_{1} – m x_{1}\)

Or directly

\(b = \dfrac{x_{2}y_{1} – x_{1}y_{2}}{x_{2} – x_{1}}\)

Worked numerical example

What to do: Apply the steps to concrete points. Suppose the points are (2, 3) and (5, 11).

1. Compute the slope:

\(m = \dfrac{11 – 3}{5 – 2} = \dfrac{8}{3}\)

2. Compute b using (2, 3):

\(b = 3 – \dfrac{8}{3}\cdot 2 = 3 – \dfrac{16}{3} = \dfrac{9}{3} – \dfrac{16}{3} = -\dfrac{7}{3}\)

3. Thus the y-intercept is

\((0, -\dfrac{7}{3})\) and the y-intercept value is \(b = -\dfrac{7}{3}\).

Follow these steps with any two non-vertical points to find the y-intercept. If the two given x-values are equal, handle the vertical-line special case described in Step 1.