Question

5,1 10,2
Derive the equation from the line y=mz

Answer 1

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 \( y = mx + b \):

1. Distribute \( \frac{1}{5} \):

\[
y - 1 = \frac{1}{5}x - 1
\]

2. 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:

\[
y = \frac{1}{5}x
\]