5. A line passes through the points (6,5) and (3, 1). Use your knowledge of slope and
different forms of equations to answer the following questions.
Part I: What is the slope of the line passing through the points (6,5) and (3, 1)?
Show your work. (2 points)
Part II: Write two point-slope equations for the line passing through the points (6,
5) and (3, 1). Show your work. (4 points: 2 points per equation)
Part III: Rewrite one of your point-slope equations from Part II in slope-intercept
form. Show your work. Be sure to show which equation you are starting with. (2
points)
I
Part IV: What is the y-intercept of the line passing through the points (6,5) and (3,
1)? Justify your answer. (2 points)

Part II: Point-slope equations of the line passing through (6,5) and (3, 1) are: [tex]y - 5 = \frac{4}{3} (x - 6)[/tex] and [tex]y - 1 = \frac{4}{3} (x - 3)[/tex]

Part III: Rewriting [tex]y - 1 = \frac{4}{3} (x - 3)[/tex] in slope-intercept form we will have: [tex]\mathbf{y = \frac{4}{3}(x) - 3}[/tex]

Part IV: The y-intercept of the line passing through (6,5) and (3,

1) is: -3

Recall:

Point-slope form equation of a line is represented with the equation, y - b = m(x - a), where, m is the slope, (a, b) is a point on the line.

Slope-intercept form equation of a line is represented with the equation, y = mx + b, where, m is the slope, and b is the y-intercept.

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

Part I: The slope of the line passing through (6,5) and (3, 1)

Let,

[tex](6,5) = (x_1, y_1)\\\\(3, 1) = (x_2, y_2)[/tex]

Substitute

[tex]Slope (m) = \frac{1 - 5}{3 - 6} = \frac{-4}{-3} \\\\\mathbf{Slope (m) = \frac{4}{3} }[/tex]

Part II: Point-slope equations of the line passing through (6,5) and (3, 1)

Substitute m = 4/3, and (a, b) = (6, 5) into y - b = m(x - a)

[tex]y - 5 = \frac{4}{3} (x - 6)[/tex] (point-slope equation)

Substitute m = 4/3, and (a, b) = (3, 1) into y - b = m(x - a)

[tex]y - 1 = \frac{4}{3} (x - 3)[/tex] (point-slope equation)

Part III: Rewriting one of the point-slope equations in slope-intercept form

Rewrite [tex]y - 1 = \frac{4}{3} (x - 3)[/tex]

[tex]y - 1 = \frac{4}{3}(x) - \frac{4}{3}(3)\\\\y - 1 = \frac{4}{3}(x) - 4\\\\y = \frac{4}{3}(x) - 4 + 1\\\\\mathbf{y = \frac{4}{3}(x) - 3}[/tex]

Part IV: The y-intercept of the line passing through (6,5) and (3,

1).

Using the equation, [tex]y = \frac{4}{3}(x) - 3[/tex],

-3 represents b in the slope-intercept form.

Therefore, the y-intercept = -3

In summary:

Part I: The slope of the line passing through (6,5) and (3, 1) is: 4/3

Part II: Point-slope equations of the line passing through (6,5) and (3, 1) are: [tex]y - 5 = \frac{4}{3} (x - 6)[/tex] and [tex]y - 1 = \frac{4}{3} (x - 3)[/tex]

Part III: Rewriting [tex]y - 1 = \frac{4}{3} (x - 3)[/tex] in slope-intercept form we will have: [tex]\mathbf{y = \frac{4}{3}(x) - 3}[/tex]

Part IV: The y-intercept of the line passing through (6,5) and (3,

1) is: -3

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