TBIL-Precal Equations of Lines (LF2)

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Section 3.2 Equations of Lines (LF2)

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Objectives

Determine an equation for a line when given two points on the line and when given the slope and one point on the line. Express these equations in slope-intercept or point-slope form and determine the slope and y-intercept of a line given an equation.

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Subsection 3.2.1 Activities

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Activity 3.2.1.

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Consider the graph of two lines.

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(a)

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Find the slope of line A.

$1$
$2$
$\frac{1}{2}$
$- 2$

Answer.

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(b)

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Find the slope of line B.

1
2
$\frac{1}{2}$
$- 2$

Answer.

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(c)

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Find the

y

-intercept of line A.

$- 2$
$- 1.5$
$1$
$3$

Answer.

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(d)

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Find the

y

-intercept of line B.

$- 2$
$- 1.5$
$1$
$3$

Answer.

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(e)

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What is the same about the two lines?

Answer.

Lines A and B have the same slope.

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(f)

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What is different about the two lines?

Answer.

Lines A and B have different

y

-intercepts.

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Remark 3.2.2.

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Notice that in Activity 3.2.1 the lines have the same slope but different

y

-intercepts. It is not enough to just know one piece of information to determine a line, you need both a slope and a point.

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Definition 3.2.3.

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Linear functions can be written in slope-intercept form

f (x) = m x + b

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where

b

is the

y

-intercept (or starting value) and

m

is the slope (or constant rate of change).

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Activity 3.2.4.

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Write the equation of each line in slope-intercept form.

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(a)

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$y = - 3 x + 1$
$y = - x + 3$
$y = - \frac{1}{3} x + 1$
$y = - \frac{1}{3} x + 3$

Answer.

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(b)

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The slope is

4

and the

y

-intercept is

(0, - 3) .

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$f (x) = 4 x - 3$
$f (x) = 3 x - 4$
$f (x) = - 4 x + 3$
$f (x) = 4 x + 3$

Answer.

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(c)

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Two points on the line are

(0, 1)

and

(2, 4) .

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$y = 2 x + 1$
$y = - \frac{3}{2} x + 4$
$y = \frac{3}{2} x + 1$
$y = \frac{3}{2} x + 4$

Answer.

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(d)

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$x$	$f (x)$
$- 2$	$- 8$
$0$	$- 2$
$1$	$1$
$4$	$10$

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$f (x) = - 3 x - 2$
$f (x) = - \frac{1}{3} x - 2$
$f (x) = 3 x + 1$
$f (x) = 3 x - 2$

Answer.

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Activity 3.2.5.

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Let’s try to write the equation of a line given two points that don’t include the

y

-intercept.

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(a)

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Plot the points

(2, 1)

and

(- 3, 4) .

Answer.

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(b)

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Find the slope of the line joining the points.

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$- \frac{5}{3}$
$- \frac{3}{5}$
$\frac{3}{5}$
$- 3$

Answer.

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(c)

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When you draw a line connecting the two points, it’s often hard to draw an accurate enough graph to determine the

y

-intercept of the line exactly. Let’s use the slope-intercept form and one of the given points to solve for the

y

-intercept. Try using the slope and one of the points on the line to solve the equation

y = m x + b

for

b .

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$2$
$\frac{11}{5}$
$\frac{5}{2}$
$3$

Answer.

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(d)

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

Answer.

y = - \frac{3}{5} x + \frac{11}{5}

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Remark 3.2.6.

It would be nice if there was another form of the equation of a line that works for any points and does not require the

y

-intercept.

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Definition 3.2.7.

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Linear functions can be written in point-slope form

y - y_{0} = m (x - x_{0})

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where

(x_{0}, y_{0})

is any point on the line and

m

is the slope.

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Activity 3.2.8.

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Write an equation of each line in point-slope form.

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(a)

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$y = \frac{1}{3} x + \frac{2}{3}$
$y - 1 = 3 (x - 1)$
$y - 1 = \frac{1}{3} (x - 1)$
$y + 2 = \frac{1}{3} (x + 2)$
$y = \frac{1}{3} (x + 2)$

Answer.

C or E

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(b)

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The slope is

4

and

(- 1, - 7)

is a point on the line.

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$y + 7 = 4 (x + 1)$
$y - 7 = 4 (x - 1)$
$y + 1 = 4 (x + 7)$
$y - 4 = 7 (x - 1)$

Answer.

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(c)

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Two points on the line are

(1, 0)

and

(2, - 4) .

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$y = - 4 x + 1$
$y - 0 = - 2 (x - 1)$
$y + 4 = - 4 (x - 2)$
$y + 4 = - 3 (x - 2)$

Answer.

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(d)

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$x$	$f (x)$
$- 2$	$- 8$
$1$	$1$
$4$	$10$

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$y + 8 = 3 (x - 2)$
$y - 1 = - \frac{1}{3} (x - 1)$
$y + 8 = - \frac{1}{3} (x + 2)$
$y - 10 = 3 (x - 4)$

Answer.

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Activity 3.2.9.

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Consider again the two points from Activity 3.2.5,

(2, 1)

and

(- 3, 4) .

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(a)

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Use point-slope form to find an equation of the line.

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$y = - \frac{3}{5} x + \frac{11}{5}$
$y - 1 = - \frac{3}{5} (x - 2)$
$y - 4 = - \frac{3}{5} (x + 3)$
$y - 2 = - \frac{3}{5} (x - 1)$

Answer.

B or C

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(b)

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Solve the point-slope form of the equation for

y

to rewrite the equation in slope-intercept form. Identify the slope and intercept of the line.

Answer.

The slope-intercept form is:

y = - \frac{3}{5} x + \frac{11}{5} .

The slope is

- \frac{3}{5}

and the

y

-intercept is

\frac{11}{5} .

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Remark 3.2.10.

Notice that it was possible to use either point to find an equation of the line in point-slope form. But, when rewritten in slope-intercept form the equation is unique.

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Activity 3.2.11.

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For each of the following lines, determine which form (point-slope or slope-intercept) would be "easier" and why. Then, write the equation of each line.

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(a)

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Answer.

Slope-intercept:

y = \frac{3}{4} x + 2

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(b)

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The slope is

- \frac{1}{2}

and

(1, - 3)

is a point on the line.

Answer.

Point-slope:

y + 3 = - \frac{1}{2} (x - 1)

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(c)

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Two points on the line are

(0, 3)

and

(2, 0) .

Answer.

Slope-intercept:

y = - \frac{3}{2} x + 3

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Remark 3.2.12.

It is always possible to use both forms to write the equation of a line and they are both valid. Although, sometimes the given information lends itself to make one form easier.

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