Lesson 2-3 ObjectiveThe student will be able to:

1) write equations using slope-intercept form.

2) identify slope and y-intercept from an equation

Important!!!

This is one of the big concepts in algebra. You need to thoroughly

understand this!

Slope – Intercept Form

y = mx + b

m represents the slope

b represents the y-intercept

Writing Equations

When asked to write an equation, you need to know two things – slope (m) and y-intercept (b).

There are three types of problems you will face.

Writing Equations – Type #1Write an equation in slope-intercept form of the

line that has a slope of 2 and a y-intercept of 6.

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

We have both!! Plug them into slope-intercept form

y = mx + b

y = 2x + 6

2

6

Write the equation of a line that has a y-intercept of -3 and a slope of -4.

1. y = -3x – 4

2. y = -4x – 3

3. y = -3x + 4

4. y = -4x + 3

Writing Equations – Type #2Write an equation of the line that has a slope of 3

and goes through the point (2,1).

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

We have to find the y-intercept!! Plug in the slope and ordered pair into

y = mx + b

1 = 3(2) + b

3

???

Writing Equations – Type #21 = 3(2) + b

Solve the equation for b

1 = 6 + b

-6 -6

-5 = b

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

y = 3x - 5

3

-5

Writing Equations – Type #3Write an equation of the line that goes through the points (-2,

1) and (4, 2). To write an equation, you need two things:

slope (m) =y – intercept (b) =

We need both!! First, we have to find the slope. Plug the points into the slope formula.

Simplify2 1

4 ( 2)m

??????

1

6m

Writing Equations – Type #3Write an equation of the line that goes through the

points (-2, 1) and (4, 2). To write an equation, you need two things:

slope (m) =y – intercept (b) =

It’s now a Type #2 problem. Pick one of the ordered pairs to plug into the equation. Which one looks easiest to use?

I’m using (4, 2) because both numbers are positive.2 = (4) + b

1

6???

1

6

Writing Equations – Type #3

2 = (4) + bSolve the equation for b

2 = + b

To write an equation, you need two things:

slope (m) =

y – intercept (b) =

1

62

32

3

11

3b

2

3

1

6 11

3

1 11

6 3y x

Write an equation of the line that goes through the points (0, 1) and (1, 4).

1. y = 3x + 4

2. y = 3x + 1

3. y = -3x + 4

4. y = -3x + 1

To find the slope and y-intercept of an equation, write the equation in slope-intercept form: y = mx + b.

Find the slope and y-intercept.

1) y = 3x – 7

y = mx + b

m = 3, b = -7

Find the slope and y-intercept.

2) y = x

y = mx + b

y = x + 0

3) y = 5

y = mx + b

y = 0x + 5

2

3m =

b = 0

2

3

2

3

m = 0b = 5

-3 -3 -3

Find the slope and y-intercept.4) 5x - 3y = 6

Write it in slope-intercept form. (y = mx + b)

5x – 3y = 6

-3y = -5x + 6

y = x - 25

3m =

b = -2

5

3

Write it in slope-intercept form. (y = mx + b)

2y + 2 = 4x

2y = 4x - 2

y = 2x - 1

Find the slope and y-intercept. 5) 2y + 2 = 4x

2 2 2

m = 2b = -1

Find the slope and y-intercept of y = -2x + 4

1. m = 2; b = 4

2. m = 4; b = 2

3. m = -2; b = 4

4. m = 4; b = -2

Graphing a Line Given a Point & Slope

• Graph a line though the point

(2, -6) with m=2/3

• Graph (2, -6)

• Count up 2 for the

rise, and to the right

3 for the run

• Plot the point, repeat, then connect

x

y

Graphing Lines• m = - ½ b = 3

• Plot y-intercept (b)

• Use the slope to

find two more

points

• Connect

x

y

Graph the Line

• (3, -3) m = undefined

x

y

POINT-SLOPE FORM

• Chapter 2-3

WarPPPPPpm-UpBELLWORK – Thurs. 10-24-13

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

2. passes through (–2, 2) and (1, 8)

ANSWER

ANSWER

1. passes through (3, 4), m = 3

y = 2x + 6

y = 3x – 5

Warm-Up

3. A carnival charges an entrance fee and a ticket fee. One person paid $27.50 and brought 5 tickets.

Another paid $45.00 and brought 12 tickets. How much will 22 tickets cost?

ANSWER $70

Example 1

Write an equation in point-slope form of the line that passes through the point (4, –3) and has a slope of 2.

y – y1 = m(x – x1) Write point-slope form.

y + 3 = 2(x – 4) Substitute 2 for m, 4 for x1, and –3 for y1.

Guided Practice

Write an equation in point-slope form of the line that passes through the point (–1, 4) and has a slope of –2.

1.

y – 4 = –2(x + 1)ANSWER

Example 2

y + 2 = (x – 3).2 3

Graph the equation

SOLUTION

Because the equation is in point-slope form, you know that the line has a slope of and passes through the point (3, –2).

2 3

Plot the point (3, –2). Find a secondpoint on the line using the slope.Draw a line through both points.

Guided Practice

–Graph the equation2.

ANSWER

y – 1 = (x – 2).

Example 3

Write an equation in point-slope form of the line shown.

SOLUTION

STEP 1

=y2 – y1

x2 – x1

m = 3 – 1 –1 – 1

=2

–2= –1

Find the slope of the line.

Example 3

Method 1 Method 2

Use (–1, 3). Use (1, 1).y – y1 = m(x – x1) y – y1 = m(x – x1)

y – 3 = –(x +1) y – 1 = –(x – 1)

STEP 2Write the equation in point-slope form. You can use either given point.

CHECK

Check that the equations are equivalent by writing them in slope-intercept form.

y – 3 = –x – 1y = –x + 2

y – 1 = –x + 1y = –x + 2

Guided Practice

Write an equation in point-slope form of the line that passes through the points (2, 3) and (4, 4).

3.

y – 3 = (x – 2) or1

2y – 4 = (x – 4)

1 2

ANSWER

Example 5

WORKING RANCH

The table shows the cost of visiting a working ranch for one day and night for different numbers of people. Can the situation be modeled by a linear equation? Explain. If possible, write an equation that gives the cost as a function of the number of people in the group.

Lesson Quiz

ANSWER y + 4 = –2(x – 6)

Write an equation in point-slope form of the line that passes through (6, –4) and has slope 2.

1.

Write an equation in point-slope form of the line that passes through (–1, –6) and (3, 10).

2.

ANSWER y + 6 = 4(x + 1) or y –10 = 4(x–3)

Lesson Quiz

A travel company offers guided rafting trips for $875 for the first three days and $235 for each additional day. Write an equation that gives the total cost (in dollars) of a rafting trip as a function of the length of the trip. Find the cost for a 7-day trip.

3.

ANSWER

C = 235t + 170, where C is total cost and t is time (in days); $1815

Write Equations and Parallel and Perpendicular Lines

Warm-Up

Are the lines parallel? Explain.

2. –x = y + 4, 3x + 3y = 5

ANSWER

ANSWER

1. y – 2 = 2x, 2x + y = 7

Yes; both slopes are –1.

No; one slope is 2 and the other is –2.

Example 1

SOLUTION

Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3x – 1.

STEP 1

Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3.

Example 1

STEP 2Find the y-intercept. Use the slope and the given point.

y = mx + b

–5 = 3(–3) + b

4 = b

Write slope-intercept form.

Substitute 3 for m, 3 for x, and 5 for y.

Solve for b.

STEP 3

Write an equation. Use y = mx + b.

y = 3x + 4 Substitute 3 for m and 4 for b.

Guided Practice

1. Write an equation of the line that passes through

(–2, 11) and is parallel to the line y = –x + 5.

y = –x + 9ANSWER

Example 2

Determine which lines, if any, are parallel or perpendicular.Line a: y = 5x – 3

Line b: x + 5y = 2

Line c: –10y – 2x = 0

SOLUTION

Find the slopes of the lines.

Line a: The equation is in slope-intercept form. The slope is 5.

Write the equations for lines b and c in slope-intercept form.

Example 2

Line b: x + 5y = 2

5y = – x + 2

Line c: –10y – 2x = 0

–10y = 2x

y = – x15xy = 2

515 +–

ANSWER

Lines b and c have slopes of – , so they are

parallel. Line a has a slope of 5, the negative reciprocal

of – , so it is perpendicular to lines b and c.

15

15

Guided Practice

Determine which lines, if any, are parallel or perpendicular.Line a: 2x + 6y = –3

Line b: y = 3x – 8

Line c: –1.5y + 4.5x = 6

ANSWER

parallel: b and c; perpendicular: a and b, a and c

Example 3

SOLUTION

Line a: 12y = –7x + 42

Line b: 11y = 16x – 52

Find the slopes of the lines. Write the equations in slope-intercept form.

The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they?

STATE FLAG

Example 3

Line a: 12y = –7x + 42

Line b: 11y = 16x – 52

y = – x + 1242 7

12

1152

y = x –1611

ANSWER

The slope of line a is – . The slope of line b is .

The two slopes are not negative reciprocals, so lines a and b are not perpendicular.

712

1611

Guided Practice

3. Is line a perpendicular to line b? Justify your answer using slopes.

Line a: 2y + x = –12

Line b: 2y = 3x – 8

ANSWER

No; the slope of line a is – , the slope of line b is . The slopes are not negative reciprocals so the lines are not perpendicular.

12

32

Example 4

SOLUTION

Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2x + 3.

STEP 1

Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is .1

2–

Example 4

STEP 2 Find the y-intercept. Use the slope and thegiven point.

Write slope-intercept form.

–5 = – (4) + b12

Substitute – for m, 4 for x, and

–5 for y.

12

y = mx + b

–3 = b Solve for b.

STEP 3 Write an equation.

y = mx + b Write slope-intercept form.

y = – x – 312 Substitute – for m and –3 for b.1

2

Guided Practice

4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4x – 7.

y = – x + 414ANSWER

Lesson Quiz

1. Write an equation of the line that passes through the point (–1, 4) and is parallel to the line y = 5x – 2.

y = 5x + 9

ANSWER

Write an equation of the line that passes through the point (–1, –1) and is perpendicular to the line y = x + 2.1

4–

2.

y = 4x + 3

ANSWER

Lesson Quiz

3. Path a, b and c are shown in the coordinate grid. Determine which paths, if any, are parallel or perpendicular. Justify your answer using slopes.

ANSWER

Paths a and b are perpendicular because their slopes, 2 and are negative reciprocals. No paths are parallel.1

2–

Graphing Lines

• Graph the line perpendicular to y = 2x + 3 that goes through the

point (-2, 3).• The slope of the line is 2 so the slope of the perpendicular line is -1/2.• m = -1/2 b = 3

Graphing Lines

• Graph the line parallel to x = -1 that goes through the point

(3, -3).

• The slope of the line is undefined (VUX) so the slope of the parallel line is undefined.

Writing Equations in Standard Form

• MFCR Lesson 2-3

Warm-Up

ANSWER

ANSWER

1. (1, 4), (6, –1)

y + 2 = 3(x + 1) or y – 7 = 3(x – 2)

y – 4 = –(x – 1) or y + 1 = –(x – 6)

2. (–1, –2), (2, 7)

Write an equation in point-slope form of the line that passes through the given points.

Example 1

To write another equivalent equation, multiply each side by 0.5.

4x – 12y = 8

To write one equivalent equation, multiply each side by 2.

SOLUTION

Write two equations in standard form that are equivalent to 2x – 6y = 4.

x – 3y = 2

Example 2

SOLUTION

y – y1 = m(x – x1)

Calculate the slope.STEP 1

–3m =1 – (–2)

1 – 2=

3–1 =

Write an equation in point-slope form. Use (1, 1).

Write point-slope form.

y – 1 = –3(x – 1) Substitute 1 for y1, 3 for m

and 1 for x1.

Write an equation in standard form of the line shown.

STEP 2

Example 2

Rewrite the equation in standard form.

3x + y = 4 Simplify. Collect variable terms on one side, constants on the other.

STEP 3

Guided Practice

Write an equation in standard form of the line through (3, –1) and (2, –3).

2.

–2x + y = –7ANSWER

Example 3

SOLUTION

Write an equation of the specified line.

The y-coordinate of the given point on the blue line is –4. This means that all points on the line have a y-coordinate of –4. An equation of the line is y = –4.

a.

The x-coordinate of the given point on the red line is 4. This means that all points on the line have an x-coordinate of 4. An equation of the line is x = 4.

b.

Blue linea. Red lineb.

Guided Practice

Write equations of the horizontal and vertical lines that pass through the given point.

3. (–8, –9)

y = –9, x = –8ANSWER

4. (13, –5)

y = –5, x = 13ANSWER

Lesson Quiz

Write an equation in standard form of the line that passes through the given point and has the given slope m or that passes through the two given points.

ANSWER 2x + y = –4

1.

(1, –6), m = –2

2. (–4, –3), (2, 9)

ANSWER –2x + y = 5