Linear Equations and Linear Equations and Slope Slope

Created by Laura Ralston Created by Laura Ralston

http://www.youtube.com/watch?v=J_U93-l5Z-w

SlopeSlope

a useful measure of the “steepness” or a useful measure of the “steepness” or “tilt” of a line “tilt” of a line

compares the vertical change (the compares the vertical change (the rise) to the horizontal change (the run) rise) to the horizontal change (the run) when moving from one point to when moving from one point to another along the lineanother along the line

typically represented by “m” because typically represented by “m” because it is the first letter of the French verb, it is the first letter of the French verb, monter monter

Formula and Graph Formula and Graph

http://www.youtube.com/watch?v=xBdo-D1RiNs

Four Possibilities of Slope Four Possibilities of Slope

Positive SlopePositive Slope• m > 0m > 0

Line “rises” from Line “rises” from left to rightleft to right

Draw graphDraw graph

Negative SlopeNegative Slope• m < 0m < 0

Line “falls” from Line “falls” from left to rightleft to right

Draw graphDraw graph

Four Possibilities of SlopeFour Possibilities of Slope

Zero SlopeZero Slope• m = 0m = 0

Line is horizontal Line is horizontal (constant)(constant)

Draw graphDraw graph

Undefined SlopeUndefined Slope• m is undefined (0 m is undefined (0

in denominator of in denominator of ratio)ratio)

Line is vertical and Line is vertical and is NOT a functionis NOT a function

Do not say “NO Do not say “NO slope”slope”

Draw graphDraw graph

Using Slope to find the Using Slope to find the equation of a line is equation of a line is IMPORTANTIMPORTANT

Linear functions can take on Linear functions can take on many forms many forms

a) Point Slope Forma) Point Slope Form

b) Slope Intercept Formb) Slope Intercept Form

c) General Form c) General Form

POINT-SLOPE FORMPOINT-SLOPE FORM

Most useful Most useful symbolic form symbolic form

Some explicit Some explicit informationinformation

Not UNIQUE since Not UNIQUE since any point can be any point can be used, but forms are used, but forms are equivalent (graphs equivalent (graphs are identical) are identical)

Can use if Can use if • slope and a point slope and a point

are known are known • or two points are or two points are

known known

y = m(x - xy = m(x - x11) + y) + y11

Where m = slope of the line Where m = slope of the line

and and

(x(x11, y, y11) is any point on the line ) is any point on the line

ExamplesExamples

Straight forward: Use the given Straight forward: Use the given conditions to write the equation for conditions to write the equation for each line. Write final answer in each line. Write final answer in slope intercept formslope intercept form• Slope =4, passing through (1, 3)Slope =4, passing through (1, 3)• Slope = Slope = , passing through (10, - 4), passing through (10, - 4)• Passing through (- 2, - 4) and (1, - 1)Passing through (- 2, - 4) and (1, - 1)• Passing through (- 2, - 5) and (6, -5)Passing through (- 2, - 5) and (6, -5)

5

3

SLOPE INTERCEPT FORMSLOPE INTERCEPT FORM

Most useful Most useful graphing formgraphing form

Some explicit Some explicit informationinformation

LIMITED in use LIMITED in use UNIQUE to the UNIQUE to the

graph graph

Can only be used if Can only be used if slope and y-slope and y-intercept are intercept are knownknown

To convert from To convert from point-slope to slope point-slope to slope intercept, apply the intercept, apply the distributive distributive property. property.

y = mx + b y = mx + b

Where m = slope of the lineWhere m = slope of the line

and and

b = y-intercept b = y-intercept

STANDARD FORMSTANDARD FORM

Every line can be Every line can be expressed in this expressed in this formform

No explicit No explicit informationinformation

Ax + By = C Ax + By = C

• where A, B, and C where A, B, and C are real numbers are real numbers with A not equal with A not equal to 0 to 0

2 SPECIAL CASES2 SPECIAL CASES

HORIZONTAL HORIZONTAL • m = 0 m = 0 • y-intercept = b y-intercept = b • all points have the all points have the

same y-coordinate same y-coordinate

• y = b or f(x) = b y = b or f(x) = b – where b is any real where b is any real

number number

VERTICAL VERTICAL • m = undefined m = undefined • no y-intercept no y-intercept • x-intercept = k x-intercept = k • all points have all points have

same x-coordinatesame x-coordinate• not a function not a function • x = k x = k

– where k is any real where k is any real number number

ExamplesExamples

Applications Applications • A business purchases a piece of A business purchases a piece of

equipment for $30,000. After 15 years, equipment for $30,000. After 15 years, the equipment will have to be replaced. the equipment will have to be replaced. Its value at that time is expected to be Its value at that time is expected to be $1,500. Write a $1,500. Write a linear equation linear equation giving giving the value, y, of the equipment in terms of the value, y, of the equipment in terms of x, the number of years after it is x, the number of years after it is purchased. What is the value of the purchased. What is the value of the equipment 5 years after it is purchased?equipment 5 years after it is purchased?

ExamplesExamples

Applications: Applications: • In 1999, there were 4076 JC Penney In 1999, there were 4076 JC Penney

stores and in 2003, there were 1078 stores and in 2003, there were 1078 JC Penney stores. Write a linear JC Penney stores. Write a linear equation that gives the number of equation that gives the number of stores in terms of the year. Let t = 9 stores in terms of the year. Let t = 9 represent 1999. Predict the number represent 1999. Predict the number of stores for the year 2008. Is your of stores for the year 2008. Is your answer reasonable? Explain. answer reasonable? Explain.

ExamplesExamples

A discount outlet is offering a 15% discount A discount outlet is offering a 15% discount on all items. Write a linear equation giving on all items. Write a linear equation giving the sale price S for an item with a list price the sale price S for an item with a list price x. x.

Dell Computers Inc pays its mircochip Dell Computers Inc pays its mircochip assembly line workers $11.50 per hour. In assembly line workers $11.50 per hour. In addition workers receive a piecework rate addition workers receive a piecework rate of $0.75 per unit. Write a linear equation of $0.75 per unit. Write a linear equation for the hourly wage W in terms of the for the hourly wage W in terms of the number of units x produced per hournumber of units x produced per hour

SPECIAL LINEAR SPECIAL LINEAR RELATIONSHIPSRELATIONSHIPS

PARALLEL : Two or more lines that PARALLEL : Two or more lines that run side by siderun side by side• never intersecting never intersecting • always same distance apart always same distance apart

• each line has the same slope meach line has the same slope m11 = = mm22

PERPENDICULAR : Two lines that PERPENDICULAR : Two lines that intersect to form 4 right angles intersect to form 4 right angles • Product of the slopes is equal to -1Product of the slopes is equal to -1

mm11mm22 = -1 = -1

Examples Examples

Passing through (-8, -10) and parallel to Passing through (-8, -10) and parallel to the line, y = - 4x + 3the line, y = - 4x + 3

Passing through (- 4, 2) and Passing through (- 4, 2) and perpendicular to the line, y = ½x + 7perpendicular to the line, y = ½x + 7

Passing through (- 2, 2) and parallel to Passing through (- 2, 2) and parallel to the line, 2x – 3y – 7 =0the line, 2x – 3y – 7 =0

Passing through (5, - 9) and Passing through (5, - 9) and perpendicular to the line, x + 7y – 12 = perpendicular to the line, x + 7y – 12 = 00