RelationsRelations

Math 314Math 314

Time FrameTime Frame

• Calculations

• Slope

• Point Slope

• Parameters

• Word Problems

CalculationsCalculations

• Sample Questions• Equations – Find x• 1 + 1 = 1 2 3 x• Find LCD = 6x • 3x + 2x = 6 • 5x = 6• x = 6/5 • x = 1.2

CalculationsCalculations

• Half a number plus a third of the same number is 11. What is the number.

• Let x = the information you know the least about. • 1x + 1x = 11 2 3 • 3x + 2x = 66• 5x = 66• x = 66/5 • x = 13.2

A Difficult Word ProblemA Difficult Word Problem

• Three people went bowling who are Sarah (S), Lucas (L) & Corey (C). Sarah bowled twice as better than Lucas. Corey bowled 15 points less than three times as better than Lucas. All together their scores were 165. How much did each person bowl.

SolutionSolution

• Sarah: 2x• Lucas x• Corey 3x – 15• 2x + x + 3x – 15 = 165• 6x – 15 = 165• 6x = 180• x = 30• One more step! Lucas 30; Sarah 60 &

Corey 75 points.

Distance Between NumbersDistance Between Numbers

• Which is closer to 11/24; ½ or 1/3?

• Put on a number line

1/3 11/24 ½

• To find the distance… subtract (highest from the lowest)

Distance Between NumbersDistance Between Numbers

• 11 – 1

24 3

= 11 – 8

24

= 3

24

• 1 – 11

2 24

= 12 – 11

24

= 1

24• Which one is closer?• 1/24

SubstitutionSubstitution

• Sometimes we look at a relationship as a formula

• Consider 2x + 8y = 16

• We have moved away from a single variable equation to a double variable equation

• It cannot be solved as is!

SubstitutionSubstitution

• If we know x = 4

• 2x + 8y = 16

• 2(4) + 8y = 16

• 8 + 8y = 16

• 8y = 8

• y = 1

Substitution Substitution • We could say that the point x = 4 and y = 1 or

(4,1) satisfies the relationship. • Ex #2. Given the relationship 5x – 7y = 210, use

proper substitution to find the coordinate (2,y)• (2,y) 5x – 7y = 210• 5(2) – 7y = 210• 10 – 7y = 210• -7y = 200• y = - 28.57• (2, -28.57)

Substitution Substitution

• Ex. #3: Given the relationship 8x + 5y = 80 (x,8)

• (x,8) 8x + 5y = 80• 8x + 5(8) = 80• 8x + 40 = 80• 8x = 40• x = 5 • (5,8)

Substitution Substitution

• Ex: #4 Given the relationship y= 3x2 – 5x – 2

• (-3,y) • (-3,y) y = 3 (-3)2 – 5 (-3) – 2• y = 3 (9) + 15 – 2• y = 40• (-3,40) • Stencil #2 (a-j)

Substitution Substitution

• Given the relationship

Linear RelationsLinear Relations

• We recall…

• Zero constant relation – horizontal

• Direct relation – through origin

• Partial relation – not through origin

• The characteristic here is the concept of a straight line – a never changing start and where it crosses the y axis

Example Example

• Line A• Line B

• We say line A has a more of a slant slope or a steeper slope than Line B

• (a of 6 compared to 2 is steeper or slope of -6 compared to -2 is steeper).

Variation RelationsVariation Relations

Name of Relation Formula Graph

Direct Relation y = mx

Partial Relation y = mx + b

Zero Variation y = b

Inverse Variation y = m

x

SlopeSlope

• What makes a slope?

Rise

Run

• We define the slope as the ratio between the rise and the run

• Slope = m = rise

run

Formula for SlopeFormula for Slope

• If we have two points (x1, y1) (x2, y2)

• Slope = m = y1 – y2 = y2 – y1

x1 – x2 x2 – x1

• Remember it is Y over X!

• Maintain order

1 2 3 4 5 6 7 8-1-2

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

A (x1, y1)

B(x2, y2)

SlopeSlope

• Consider two points

A (5,4), B (2, 1) what is the slope?

Calculating SlopeCalculating Slope

• Slope = m = y1 – y2 = y2 – y1

x1 – x2 x2 – x1

(5, 4) (2, 1) 4 - 1 5 - 2 3 3m = 1

(x1,y1) (x2,y2)

Ex # 2Ex # 2 A = (-4, 2) B=(2, -4)

(x1,y1) (x2,y2) -4 – 2

2 - - 4

- 6

6

m = -1

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y (4, 5)

(1, 1)

3

414

15slope

Ex #3Ex #3

(x2,y2)

(x1,y1)

21

21

xx

yyslope

Understanding the SlopeUnderstanding the Slope

• If m or the slope is 2 this means a rise of 2 and a run of 1 (2 can be written as 2 ) 1• If m = - 5, this means a rise of -5 and right

1• If m= -2 this means rise of -2 right 3 3 • Rise can go up or down, run must go right

Consider y = 2x + 3Consider y = 2x + 3

• What is the slope, y intercept, rise & run?

• We can write the slope 2 as a fraction 2

1

• We have a y intercept of 3

• This means rise of 2, run of 1

• Look at previous slide for slope of 4/3

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Ex#1: y=2x+3Ex#1: y=2x+3

0,3

(1,5)

Question: Draw this line

What is the y intercept?

What is the slope

What does the slope mean?

Where can you plot the y intercept?

Up 2, Right 1

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

(-4, 2) (2,2)

06

0

)4(2

22

slopeIf a line//If a line//xx--

axisaxis

slope = 0slope = 0

ExampleExample

• What do you think the slope will be; calculate it.

1 2 3 4 5-1-2-3-4-5

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

(2,-3)

(2,2)0

522

32

slope

If a line // If a line // yy--axis:axis:

slope is slope is undefinedundefined

ExampleExample

zero!

In Search of the EquationIn Search of the Equation

• We have seen that the linear relation or function is defined by two main characteristics or parameters

• A parameter are characteristics or how we describe something

• If we consider humans, a parameter would be gender. (We have males & females). There can be many other parameters (blonde hair, blue eyes, etc.)

In Search of the Equation NotesIn Search of the Equation Notes

• The parameters we are concerned with are…

• Slope = m = the slope of the line

• y intercept = b = where the line touches or crosses the y axis (It can always be found by replacing x = 0)

• x intercept = where on the graph the line touches or crosses the x axis. (let y = 0)

In Search of the Equation NotesIn Search of the Equation Notes

• We stated in standard form the equation for all linear functions by y = mx + b. Recall…

• y is the Dependent Variable (DV)• m is the slope• x is the Independent Variable (IV)• b is the y intercept parameter• The key is going to be finding the specific

parameters.

General FormGeneral Form

• You will also be asked to write in general form

• General Form Ax + By + C = 0

• A must be positive

• Maintain order x, y, number = 0

• No fractions

General Form PracticeGeneral Form Practice

• Consider y = 6x – 56

• -6x + y + 56 = 0

• 6x – y – 56 = 0

Standard & General FormStandard & General FormExample #1Example #1

• State the equation in standard and general form. • Consider find the equation of the linear function

with slope of m and passing through (x, y).• m = -6 (-2, -3)• (-2, -3) -3 = -6 (-2) + b• -3 = 12 + b• -15 = b• b = -15

Example #1 Solution Con’tExample #1 Solution Con’t

• y = -6x – 15 (Standard)

• Now put this in general form

• 6x + y +15 = 0 (General)

Standard & General Form Ex. #2Standard & General Form Ex. #2• m = -2 (5, - 3) 3 • -3 = (-2) (5) + b 3• -3 = -10 + b 3• -9 = -10 + 3b• 1 = 3b• b = 1/3• y = -2 x + 1 (SF)

3 3

• Now General form• Get rid of the

fractions; how?Given y = -2 x + 1

3 3…

Anything times the bottom gives you the top

• 3y = -2x + 1• 2x + 3y – 1 = 0

Standard and General Form Ex #3 Standard and General Form Ex #3

• m = 4

5 (-1, -1)• -1 = 4 x + b

5• -5y = -4x + 5b• 5 (-1) = 4 (-1) + 5b• -5 = -4 + 5b • -1 = 5b • b = -1/5

• y = 4x – 1

5 5• 5 x – 1/5 (standard

form)• 5y = 4x – 1• -4x +5y + 1 = 0 • 4x – 5y – 1 = 0

(general form)

The Point Slope Method Con’tThe Point Slope Method Con’t

• Consider, find the equation of the linear function with slope 6 and passing through (9 – 2).

• Take a look at what we know based on this question.

• m = 6

• x = 9

• y = -2

Finding the Equation in Standard Finding the Equation in Standard FormForm

• We know y = mx + b• We already know y = 6x + b• What we do not know is the b parameter or the y

intercept• We will substitute the point • (9, -2) - 2 = (6) (9) + b• -2 = 54 + b• -56 = b• b = - 56• y = 6x – 56 (this is Standard Form)• Standard from is always y = mx + b (the + b part can be

negative… ). You must have the y = on the left hand sides and everything else on the right hand side.

General FormGeneral Form

• In standard form y = 6x – 56

• In general form -6x + y + 56 = 0

• 6x – y – 56 = 0

Example #1 8a on StencilExample #1 8a on Stencil

• In the following situations, identify the dependent and independent variables and state the linear relations

• Little Billy rents a car for five days and pays $287.98. Little Sally rents a car for 26 days and pays $1195.39.

• D.V $ Money $

• I.V. # of days

Example #1 Soln Con’tExample #1 Soln Con’t

• Try and figure out the equation

• y = mx + b (you want 1 unknown)

• (5, 287.98) (26, 1195.39)

• m = (287.98 – 1195.39)

5 – 26

• m = 43.21

Unknown Unknown

Example #1 Soln Con’tExample #1 Soln Con’t

• Solve for b…

• y = mx + b

• (5, 287.98) 287.98 = 43.21 (5) + b

• 287.98 = 216.05 + b

• 71.93 = b

• b = 71.93

• y = 43.21x + 71.93

Example #2 8 b on StencilExample #2 8 b on Stencil

• A company charges $62.25 per day plus a fixed cost to rent equipment. Little Billy pays $1264.92 for 19 days.

• I.V. # of days

• D.V. Money

• m = 62.25

Example #2 8a Soln Example #2 8a Soln

• y = mx + b

• (19, 1264.92) 1264.92 = 62.25 (19) + b

• 1264.92 = 1182.75 + b

• 82.17 = b

• b = 82.17

• y = 62.25x + 82.17

Solutions 8 c, d, eSolutions 8 c, d, e

• 8c) IV # of days; DV $

• y = 47.15x + 97.79

• 8d) IV # of days; DV $

• y = 89.97x + 35.22

• 8e) IV # of days DV $

• y= 45.13x + 92.16

Homework HelpHomework Help

• What is the value of x given• 3 = 1 + 1 4 2 x• Eventually, x on the left side, number on the

right side• 3 – 1 = 1 4 2 x• 6x – 4x = 8• -2x = 8• x = -4

• Important step to understand

Homework HelpHomework Help

• What is the opposite of ½ ? • Answer is –½ • If asked what is the opposite of subtracting two

fractions… i.e. ¼ - ½ , find the answer (lowest common denominator and then reverse the sign.

• When told price increases 10% each year… calculate new price after year 1 and then multiply that number by .1 again to calculate price increase for year 2. For example, you have $100 and increases 10%. After year 1 $110 (100 x .1 + 100) & after year two $121 (110 x .1 + 110).