To derive the equation of the line in the form \( y = mx + b \) from the given points \((5, 1)\) and \((10, 2)\), we first need to determine the slope \( m \) of the line passing through these points.
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the points:
\[ (x_1, y_1) = (5, 1) \] \[ (x_2, y_2) = (10, 2) \]
we have:
\[ m = \frac{2 - 1}{10 - 5} = \frac{1}{5} \]
Now that we have the slope \( m = \frac{1}{5} \), we can use the point-slope form of the line \( y - y_1 = m(x - x_1) \) to find the equation of the line. We can use either point; let's use the point \((5, 1)\):
\[ y - 1 = \frac{1}{5}(x - 5) \]
Now, we will simplify this to the slope-intercept form \( y = mx + b \):
- Distribute \( \frac{1}{5} \):
\[ y - 1 = \frac{1}{5}x - 1 \]
- Add 1 to both sides:
\[ y = \frac{1}{5}x + 0 \]
So the equation of the line is:
\[ y = \frac{1}{5}x \]
This equation fits in the form \( y = mx \), where \( m = \frac{1}{5} \). Therefore, the derived equation from the given points is:
\[ y = \frac{1}{5}x \]
17 answers