The ratio of vertical change to horizontal change of a line is defined by point gradient. A gradient is also known as a derivative. The gradient of a line is m =
\(\begin{array}{l}\frac{rise}{run}\end{array} \)
.
m = point gradient of a line.
The formula is:
\[\large m=\frac{y-y_{1}}{x-x_{1}}\]
Solved Example
Question: Find the equation of the straight line that has slope m = 4 and passes through the point (–1, –5).
Solution:
Given,
\(\begin{array}{l}\left( x_{1},y_{1}\right )\end{array} \)
= (–1, –5)
m = 4
Point Gradient Formula:
\(\begin{array}{l}y – y_1 = m \left( x – x_1 \right )\end{array} \)
\(\begin{array}{l}y – (-5) = 4 \left (x – (-1) \right )\end{array} \)
\(\begin{array}{l}y + 5 = 4 \left ( x+1 \right )\end{array} \)
y = 4x + 4 – 5y = 4x – 1
This is the required equation.
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