A line passes through the points [tex]\((66, 33)\)[/tex] and [tex]\((99, 51)\)[/tex].

a.) What is the slope of this line? (Write your answer as a simplified fraction)
[tex]\(\square\)[/tex]

b.) Write the equation of this line in point-slope form:
[tex]\(\square\)[/tex]

c.) Write the equation of this line in slope-intercept form:
[tex]\(\square\)[/tex]

d.) Does this line pass through the point [tex]\((-11, -8)\)[/tex]?
A. No
B. Yes



Answer :

Let's solve this step by step.

1. Calculate the slope of the line passing through (66,33) and (99,51)

The formula for the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:

[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

Substituting the given points [tex]\((66, 33)\)[/tex] and [tex]\((99, 51)\)[/tex]:

[tex]\[ m = \frac{51 - 33}{99 - 66} = \frac{18}{33} = \frac{6}{11} \][/tex]

So, the slope [tex]\(m\)[/tex] is [tex]\( \boxed{\frac{6}{11}} \)[/tex].

2. Write the equation of this line in point-slope form

The point-slope form of the equation of a line is given by:

[tex]\[ y - y_1 = m(x - x_1) \][/tex]

Using one of the given points [tex]\((66, 33)\)[/tex] and the slope [tex]\(\frac{6}{11}\)[/tex]:

[tex]\[ y - 33 = \frac{6}{11}(x - 66) \][/tex]

Thus, the point-slope form is [tex]\( \boxed{y - 33 = \frac{6}{11}(x - 66)} \)[/tex].

3. Write the equation of this line in slope-intercept form

The slope-intercept form of a line is:

[tex]\[ y = mx + b \][/tex]

To find [tex]\(b\)[/tex], we use one of the given points and the slope. Again using the point [tex]\((66, 33)\)[/tex] and the slope [tex]\(\frac{6}{11}\)[/tex]:

Substitute [tex]\(x = 66\)[/tex], [tex]\(y = 33\)[/tex], and [tex]\(m = \frac{6}{11}\)[/tex] into the slope-intercept form:

[tex]\[ 33 = \frac{6}{11}(66) + b \][/tex]

[tex]\[ 33 = 6 \times 6 + b \][/tex]

[tex]\[ 33 = 36 + b \][/tex]

Solving for [tex]\(b\)[/tex], we get:

[tex]\[ b = 33 - 36 = -3 \][/tex]

Thus, the slope-intercept form is [tex]\( \boxed{y = \frac{6}{11}x - 3} \)[/tex].

4. Determine if the line passes through the point [tex]\((-11, -8)\)[/tex]

We need to check if the point [tex]\((-11, -8)\)[/tex] satisfies the equation of the line:

Using the slope-intercept form [tex]\(y = \frac{6}{11}x - 3\)[/tex], substitute [tex]\(x = -11\)[/tex] and see if [tex]\(y\)[/tex] equals [tex]\(-8\)[/tex]:

[tex]\[ y = \frac{6}{11}(-11) - 3 = -6 - 3 = -9 \][/tex]

Since [tex]\(-9 \neq -8\)[/tex], the point [tex]\((-11, -8)\)[/tex] does not lie on the line.

Therefore, the answer is [tex]\( \boxed{\text{A. No}} \)[/tex].