What is the equation of the line containing the points (7,3), (14,6), and (21,9)?

A. [tex]\( y = x - 4 \)[/tex]
B. [tex]\( y = \frac{1}{7} x \)[/tex]
C. [tex]\( y = \frac{3}{7} x \)[/tex]



Answer :

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).