Chapter 5: Writing Linear Equations

I. Writing Linear Equations in Slope-Intercept Form

OBJECTIVE: use the slope-intercept form to write an equation of a line

STANDARDS: 2.1, 2.2, 2.4, 2.5, 2.8

A. Knowing the slope and intercept

1. m = -2, b = 10

2. m = ¾, b = -4

B. Looking at a graph

II. Writing Linear Equations Given the slope and a point

OBJECTIVE: use slope and any point on a line to write an equation of the line

STANDARDS: 2.1, 2.2, 2.4, 2.5, 2.8

A. Given a point slope

1. point (6, -3) m = -2

2. point (-3, 0) m = ⅓

B. Parallel Lines (given first equation and a point)

1. y = -3x – 2 through the point (3, -4)

2. y = ⅔x – 2 through the point (-2, 1)

C. Examples of Real Life Situations

In 1990 the population of South Carolina was approximately 3,486,000. During the next five years, the population increased by approximately 37,400 people per year.

write an equation to model the population P of South Carolina in terms of t, the number of years since 1990.

b. estimate the population the South Carolina in 1996

2. You borrow $40 from your sister. To repay the loan, you pay $5 a week. Write a linear equation to model the situation.

3. You are designing a calendar as a fund-raising project for our Biology club. The cost of printing is $500, plus $2.50 per calendar. Write a linear equation to model the situation for the total cost, C, of printing x number of calendars.

III. Writing Linear Equations Given Two Points

OBJECTIVE: write an equation of a line given two points on the line

STANDARDS: 2.1, 2.2, 2.4, 2.5, 2.8

A. Steps

1. Find the slope

2. Find the y-intercept

3. Write an equation

B. Example

1. (1, 6) (3, -4)

2. (-3, 1)(5,5)

C. Perpendicular Lines – slopes are negative reciprocals of each other)

1. (0, 4)(2,0) and (-6, 1)(0,4)

2. (-4, -3) (2,0) and (-6,1)(-4, -3)

3. (2, 3) (-1, 2) and (3,0) (2, 3)

D. Real Life Situations

1. While working at an archaeological dig, you find an upper leg bone (femur) that belonged to an adult human male. The bone is 43 centimeters long. In humans, femur length is linearly related to height. To estimate the height of the person, you measure the femur and height of two complete adult male skeletons found at the same excavation.

a. person 1: 40 cm femur, 162 cm height

b. person 2: 45 cm femur, 173 cm height

2. For science class, you need to know the Celsius equivalent of a normal room temperature of 700 F. You know that Celsius temperature is linearly related to Fahrenheit temperature. To estimate the Celsius equivalent, you use what you know about the Fahrenheit and Celsius temperatures of the freezing and boiling points of water.

a. Freezing point: 320 F, 00 C

b. Boiling point: 2120 F, 1000 C

Estimate the Celsius equivalent of normal room temperature.

IV. Fitting a Line to Data

OBJECTIVES: find a linear equation that approximates a set of data points; determine whether the slopes correlation

STANDARDS: 2.1, 2.2, 2.4, 2.5, 2.8, 2.11

A. Example

You are studying the way a tadpole turns into a frog. You collect data to make a table that shows the ages and lengths of the tails of 8 tadpoles. Approximate the best-fitting line for your data. Write an equation of your line.

AGE (day)Length of Tail (mm)

514

215

121

103

312

The winning Olympic times for the women’s 100 meter run from 1948 to 1996 are shown in the table. Approximate the best-fitting line for these times. Write an equation of your line.

Olympic YearWinning Time

194811.9 s

195211.5 s

195611.5 s

196011.0 s

196411.4 s

196811.0 s

197211.1 s

197611.1 s

198011.1 s

198411.0 s

198810.5 s

199210.8 s

199610.9 s

V. Point Slope Form

OBJECTIVE: use the point-slope form to write an equation of a line

STANDARDS: 2.1, 2.2, ,2.4, 2.5, 2.8

Point Slope form – the equation of the nonvertical line that passes through a given point (x, y) with a slope of m is y- y1 = m(x – x1)

B. Examples

1. (-3, 6) m = -2

2. (-2, 3) slope = 3

3. (3,6) and (1, -2)

4. A crate can hold up to 30 oranges. The crate with 16 oranges weighs 8.8 lbs. Assuming each orange weighs 0.4 lbs, write a linear model to represent the weight of the crate with x oranges. How much will 30 oranges weighs?

VI. Standard Form of a Linear Equation

OBJECTIVE: write a linear equation in standard form

STANDARDS: 2.1, 2.2, 2.4, 2.5, 2.8

A. Standard form – a linear equation written in the form Ax + By = C, where A, B, and C are integers and A and B are not both zero.

B. Examples

1. Are these equivalent equations?

y = 4/3x – 2 and -4x + 3y = -6

2. y = 2/5x – 3

3. (-4, 3) slope = -2

4. (-5, 1) m = ¾

5. (-2, 3) slope = 0

6. (-3, -1) slope = undefined

7. (3,0) (-5, 3)

8. You are in charge of buying the hamburger and chicken for a barbecue. The hamburger costs $2 per pound and the chicken $3 per pound. You have $30 to spend. Write the linear model in standard form.

VII. Predicting with Linear Models

OBJECTIVE: determine whether a linear model is appropriate

STANDARDS: 2.1, 2.2, 2.4, 2.5, 2.8, 2.11

A. Vocabulary

1. Linear interpolation – a method of estimating the coordinates of a point that lies BETWEEN two given data points.

2. Linear extrapolation – a method of estimating the coordinates of a point that lies to the RIGHT or LEFT of all the given data points.

B. Examples

1. The amount (in millions of dollars) spent on advertising in broadcast television and on the Internet from 1995 through 2001 is given in the table. Which data are better modeled with a linear model?

Year1995199719992001

Broadcast32,72036,89341,23046,140

Internet24130015,00046,000