Relations Math 314

Time Frame Slope Point Slope Parameters Word Problems

Substitution 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!

Substitution If we know x = 4 2x + 8y = 16 2(4) + 8y = y = 16 8y = 8 y = 1

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 = – 7y = y = 200 y = (2, )

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

Linear 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 Line A Line B We say line A has a more of a slant slope or a steeper slope (6 compared to 2 is steeper or -6 compared to -2 is steeper).

Variation Relations Name of RelationFormulaGraph Direct Relationy = mx Partial Relationy = mx + b Zero Variationy = b Inverse Variationy = m x

Slope 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 Slope If we have two points (x 1, y 1 ) (x 2, y 2 ) Slope = m = y 1 – y 2 = y 2 – y 1 x 1 – x 2 x 2 – x 1 Remember it is Y over X! Maintain order

A (x 1, y 1 ) B(x 2, y 2 ) Slope Consider two points A (5,4), B (2, 1) what is the slope?

Calculating Slope Slope = m = y 1 – y 2 = y 2 – y 1 x 1 – x 2 x 2 – x 1 (5, 4) (2, 1) m = 1 (x 1,y 1 ) (x 2,y 2 )

Ex # 2 A = (-4, 2) B=(2, -4) (x 1,y 1 ) (x 2,y 2 ) -4 – m = -1

(4, 5) (1, 1) Ex #3 (x 2,y 2 ) (x 1,y 1 )

Understanding 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 + 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

Ex#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

(-4, 2) (2,2) If a line//x-axis slope = 0 Example What do you think the slope will be; calculate it.

(2,-3) (2,2) If a line // y-axis: slope is undefined Example zero!

In 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 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 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 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 Practice Consider y = 6x – 56 -6x + y + 56 = 0 6x – y – 56 = 0

Standard & General Form Example #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 Cont y = -6x – 15 (Standard) Now put this in general form 6x + y +15 = 0 (General)

Standard & General Form Ex. #2 m = -2 (5, - 3) 3 -3 = (-2) (5) + b 3 -3 = b 3 -9 = b 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 … Anything times the bottom gives you the top 3y = -2x + 1 2x + 3y – 1 = 0

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 = b -1 = 5b b = -1/5 y = 4x – x – 1/5 (standard form) 5y = 4x – 1 -4x +5y + 1 = 0 4x – 5y – 1 = 0 (general form)

The Point Slope Method Cont 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 Form 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 Form In standard form y = 6x – 56 In general form -6x + y + 56 = 0 6x – y – 56 = 0

Example #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 $ Little Sally rents a car for 26 days and pays $ D.V $ Money $ I.V. # of days

Example #1 Soln Cont Try and figure out the equation y = mx + b (you want 1 unknown) (5, ) (26, ) m = ( – ) 5 – 26 m = Unknown

Example #1 Soln Cont Solve for b… y = mx + b (5, ) = (5) + b = b = b b = y = 43.21x

Example #2 8 b on Stencil A company charges $62.25 per day plus a fixed cost to rent equipment. Little Billy pays $ for 19 days. I.V. # of days D.V. Money m = 62.25

Example #2 8a Soln y = mx + b (19, ) = (19) + b = b = b b = y = 62.25x

Solutions 8 c, d, e 8c) IV # of days; DV $ y = 47.15x d) IV # of days; DV $ y = 89.97x e) IV # of days DV $ y= 45.13x

Homework Help What is the value of x given 3 = x Eventually, x on the left side, number on the right side 3 – 1 = x 6x – 4x = 8 -2x = 8 x = -4 Important step to understand

Homework 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 ) & after year two $121 (110 x ).