Linear Equations and Slope Created by Laura Ralston

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Slope b a useful measure of the “steepness” or “tilt” of a line b compares the vertical change (the rise) to the horizontal change (the run) when moving from one point to another along the line b typically represented by “m” because it is the first letter of the French verb, monter

Formula and Graph D1RiNs

Four Possibilities of Slope b Positive Slope m > 0 m > 0 b Line “rises” from left to right b Draw graph b Negative Slope m < 0 b Line “falls” from left to right b Draw graph

Four Possibilities of Slope b Zero Slope m = 0 m = 0 b Line is horizontal (constant) b Draw graph b Undefined Slope m is undefined (0 in denominator of ratio) b Line is vertical and is NOT a function b Do not say “NO slope” b Draw graph

Using Slope to find the equation of a line is IMPORTANT Using Slope to find the equation of a line is IMPORTANT Linear functions can take on many forms a) Point Slope Form b) Slope Intercept Form c) General Form

POINT-SLOPE FORM b Most useful symbolic form b Some explicit information b Not UNIQUE since any point can be used, but forms are equivalent (graphs are identical) b Can use if slope and a point are known or two points are known

y = m(x - x 1 ) + y 1 Where m = slope of the line and (x 1, y 1 ) is any point on the line

Examples b Straight forward: Use the given conditions to write the equation for each line. Write final answer in slope intercept form Slope =4, passing through (1, 3)Slope =4, passing through (1, 3) Slope =, passing through (10, - 4)Slope =, 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)

SLOPE INTERCEPT FORM b Most useful graphing form b Some explicit information b LIMITED in use b UNIQUE to the graph b Can only be used if slope and y-intercept are known b To convert from point- slope to slope intercept, apply the distributive property.

y = mx + b Where m = slope of the line and b = y-intercept

STANDARD FORM b Every line can be expressed in this form b No explicit information b Ax + By = C where A, B, and C are real numbers with A not equal to 0

2 SPECIAL CASES b HORIZONTAL m = 0m = 0 y-intercept = by-intercept = b all points have the same y-coordinateall points have the same y-coordinate y = b or f(x) = by = b or f(x) = b –where b is any real number b VERTICAL m = undefined no y-intercept x-intercept = k all points have same x- coordinate not a function x = k –where k is any real number

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

Examples b Applications: In 1999, there were 4076 JC Penney stores and in 2003, there were 1078 JC Penney stores. Write a linear equation that gives the number of stores in terms of the year. Let t = 9 represent Predict the number of stores for the year Is your answer reasonable? Explain.In 1999, there were 4076 JC Penney stores and in 2003, there were 1078 JC Penney stores. Write a linear equation that gives the number of stores in terms of the year. Let t = 9 represent Predict the number of stores for the year Is your answer reasonable? Explain.

Examples b A discount outlet is offering a 15% discount on all items. Write a linear equation giving the sale price S for an item with a list price x. b Dell Computers Inc pays its mircochip assembly line workers $11.50 per hour. In addition workers receive a piecework rate of $0.75 per unit. Write a linear equation for the hourly wage W in terms of the number of units x produced per hour

SPECIAL LINEAR RELATIONSHIPS b PARALLEL : Two or more lines that run side by side never intersectingnever intersecting always same distance apartalways same distance apart each line has the same slope m 1 = m 2each line has the same slope m 1 = m 2

b PERPENDICULAR : Two lines that intersect to form 4 right angles Product of the slopes is equal to -1Product of the slopes is equal to -1 m 1 m 2 = -1

Examples b Passing through (-8, -10) and parallel to the line, y = - 4x + 3 b Passing through (- 4, 2) and perpendicular to the line, y = ½x + 7 b Passing through (- 2, 2) and parallel to the line, 2x – 3y – 7 =0 b Passing through (5, - 9) and perpendicular to the line, x + 7y – 12 = 0