Certainly! To show that if the points [tex]\((x, y)\)[/tex], [tex]\((0, c)\)[/tex], and [tex]\(\left(-\frac{c}{m}, 0\right)\)[/tex] are collinear, then the equation [tex]\(y = mx + c\)[/tex] holds.
Step-by-Step Solution:
1. Concept of Collinearity:
Points are collinear if they lie on the same straight line. If three points are collinear, the slope between any two pairs of these points must be the same.
2. Determine Slopes:
Calculate the slope between each pair of points. The first pair of points we consider is [tex]\((x, y)\)[/tex] and [tex]\((0, c)\)[/tex]. The slope [tex]\(m_{1,2}\)[/tex] between these points is given by:
[tex]\[
m_{1,2} = \frac{c - y}{0 - x} = \frac{c - y}{-x} = \frac{y - c}{x}
\][/tex]
Next, consider the pair [tex]\((0, c)\)[/tex] and [tex]\(\left(-\frac{c}{m}, 0\right)\)[/tex]. The slope [tex]\(m_{2,3}\)[/tex] between these points is:
[tex]\[
m_{2,3} = \frac{0 - c}{-\frac{c}{m} - 0} = \frac{-c}{-\frac{c}{m}} = \frac{-c \cdot m}{-c} = m
\][/tex]
3. Set Slopes Equal:
For the points to be collinear, these slopes must be equal:
[tex]\[
\frac{y - c}{x} = m
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
Rearranging the above equation to isolate [tex]\(y\)[/tex]:
[tex]\[
y - c = mx
\][/tex]
Finally, add [tex]\(c\)[/tex] to both sides:
[tex]\[
y = mx + c
\][/tex]
Therefore, the equation [tex]\(y = mx + c\)[/tex] is proven. This means that if the points [tex]\((x, y)\)[/tex], [tex]\((0, c)\)[/tex], and [tex]\(\left(-\frac{c}{m}, 0\right)\)[/tex] are collinear, then [tex]\(y\)[/tex] must indeed satisfy the equation [tex]\(y = mx + c\)[/tex].
