To determine the equation of the line that passes through the points [tex]\((7, 3)\)[/tex], [tex]\((14, 6)\)[/tex], and [tex]\((21, 9)\)[/tex], we need to find the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]) of the line.
Step 1: Calculate the slope [tex]\(m\)[/tex]
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Let’s use the points [tex]\((7, 3)\)[/tex] and [tex]\((14, 6)\)[/tex]:
[tex]\[
m = \frac{6 - 3}{14 - 7} = \frac{3}{7}
\][/tex]
The slope [tex]\(m\)[/tex] is [tex]\(\frac{3}{7}\)[/tex].
Step 2: Calculate the y-intercept [tex]\(b\)[/tex]
Once we have the slope, we can find the y-intercept [tex]\(b\)[/tex] using the equation of a line [tex]\( y = mx + b \)[/tex]. To do this, we can substitute one of the points and the slope into the line equation and solve for [tex]\( b \)[/tex].
Using the point [tex]\((7, 3)\)[/tex]:
[tex]\[
3 = \left(\frac{3}{7}\right) \cdot 7 + b
\][/tex]
[tex]\[
3 = 3 + b
\][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[
b = 3 - 3 = 0
\][/tex]
So, the y-intercept [tex]\( b \)[/tex] is [tex]\( 0 \)[/tex].
Step 3: Write the equation of the line
The equation of the line in slope-intercept form [tex]\( y = mx + b \)[/tex] is:
[tex]\[
y = \left(\frac{3}{7}\right)x + 0
\][/tex]
or simply:
[tex]\[
y = \frac{3}{7}x
\][/tex]
Therefore, the equation of the line that passes through the points [tex]\((7, 3)\)[/tex], [tex]\((14, 6)\)[/tex], and [tex]\((21, 9)\)[/tex] is:
[tex]\[ \boxed{y = \frac{3}{7}x} \][/tex]
This corresponds to option (C).
