This equation shows the definition of the gradient of a straight line.
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( y \) | This is a secondary symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( m \) | This is the symbol for the gradient of a straight line. It is a ratio between a change in the input, and a change in the output. |
This is by definition of the line gradient:
The symbol, \(\nabla\), represents the gradient of a multivariable function \( \htmlClass{sdt-0000000008}{u} \), that takes in multiple arguments. The symbol signifies a vector operator that operates on a scalar function, resulting in partial derivates of the function.
For example if we have a function \( \htmlClass{sdt-0000000008}{u} \) : (\( \htmlClass{sdt-0000000003}{x} \), \( \htmlClass{sdt-0000000017}{y} \)) \(\rightarrow\) \( \htmlClass{sdt-0000000043}{z} \), then its gradient is \[\nabla(\htmlClass{sdt-0000000008}{u}) = (\frac{\delta \htmlClass{sdt-0000000043}{z}}{\delta \htmlClass{sdt-0000000003}{x}}, \frac{\delta \htmlClass{sdt-0000000043}{z}}{\delta \htmlClass{sdt-0000000017}{y}})\]
Finding the gradient of the line in the following figure:
Let's consider the ends of the lines as displayed, (-3, -5) and (2, 5)
\(\Delta \htmlClass{sdt-0000000017}{y}\) will be the change in the \(\htmlClass{sdt-0000000017}{y}\) direction. This will be \(5 -- 5\).
\(\Delta \htmlClass{sdt-0000000003}{x}\) will be the change in the \(\htmlClass{sdt-0000000003}{x}\) direction. This will be \(2--3\).
Therefore, the gradient will be.
\[ \htmlClass{sdt-0000000131}{X}lineGradient = \frac{5--5}{2--3} = \frac{10}{5} = 2 \]