Section 3.5. Equations of Lines

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1 Section 3.5 Equations of Lines

2 Learning objectives Use slope-intercept form to write an equation of a line Use slope-intercept form to graph a linear equation Use the point-slope form to find an equation of a line given its slope and a point on the line Use point-slope form to find an equation of a line given two points on the line Use point-slope form to solve word problems Vocabulary: standard form; slope-intercept form; point-slope form

3 Slope-intercept form As we learned in Section 3.2, the standard form of a linear equation with two variables is as follows: If A, B, and C are real numbers, then Ax + By = C As we learned in Section 3.4, if we solve a linear equation with two variables for y, we end up with an equation: y = mx + b where m is slope of the line & b a real number This represents the linear equation in slope-intercept form, where b is the y-intercept of the line with point (0,b)

4 Consider y = 3x + 1 This is an equation in slope-intercept form: m = 3 b = 1, so the y-intercept of the line is at (0,1) y = 3x ,

5 Find an equation with y-intercept (0,-3) and slope of 1/4 Slope = m = ¼ Y-intercept = b = -3 The slope-intercept form of equation is y = mx + b Therefore y = (1/4)x + (-3) y = 1/4x - 3 y = 1/4x

6 Write an equation of the line with each given slope, m, and y-intercept, (0,b) m = -9, b = 4 m = -2/3, b = 7 m = 0, b = ½ m = -5/2, b =15 1/2

7 Using the Slope-intercept form to graph an equation Graph y = 3/5x 2 This equation is written in slope-intercept form m = 3/5 y-intercept : (0,-2) y = 3/5x , First: put a point on (0,-2) Then: count up 3 spaces from (0,-2), then count right 5 spaces to (5,1) Put a point on (5,1) and draw your line , -2

8 Use slope-intercept form to graph the equation 4x + y = 1 Get the equation in slope-intercept form by subtracting 4x from each side So y = -4x + 1 Slope m = -4 Y-intercept b = (0,1) Put your first point on (0,1) Then count down 4 then count over 1 to the right Put your second point at (1,-3) and draw your line y = -4x , , -3-4

9 Use slope-intercept form to graph: y = 1/2x 3 y = -1/4x + 2 5x + 2y = 10

10 Writing an equation given slope and point What if we are not given an equation, just the slope m and a point on the line (x1,y1)? We then pretend there is another point on the line and call it plain old (x,y) By the definition of slope m = rise/run = (y y1)/(x x1) Multiply both sides by (x x1) You are left with (y y1) = m(x x1) This is the point-slope form of the equation of a line

11 Find the equation of a line with slope -2 that passes through (-1,5) If m = -2, x1 = -1 and y1 = 5; Point-slope form is: y 5 = -2(x (-1)) Or: y 5 = -2(x + 1) Distribute the equation: y 5 = -2x 2 Solve for y: y = -2x + 3 Slope intercept form is: y = -2x + 3; y-intercept (0,3) Now add 2x to both sides: 2x + y = 3 This is Standard form.

12 Write the equation in point-slope form, then write it in standard form m = 4; (10,5) m = -7/9; (5,2) m = -6; (-8,-10)

13 Writing an equation given two points Suppose we are only given two points of a line (x1,y1) and (x2,y2)? We can find the slope m = rise/run = (y2-y1)/(x2-x1) Once we get the slope, then we can write the equation in point-slope form using one of the two points (x1,y1) and m Point-slope form: (y y1) = m(x x1) Once we obtain point-slope form, we can then derive slope-intercept form: y = mx m(x1) + y1

14 Find the equation of the line through (2,5) and (-3,4) First, find m: (y2-y1)/(x2-x1) = (4-5)/(-3 2) = 1/5 Then, use m = 1/5 and the point (2,5) to write an equation in point-slope form: y y1 = m(x x1) y 5 = 1/5(x 2) Now, distribute, then add 5 to each side: y = 1/5x 2/5 + 5 = 1/5x + 4 3/5 y = 1/5x + 4 3/5

15 Find the equation of the line passing through each of two points: (-7,-4) and (0,5) (3,7) and (-2,-6) (9,-9) and (6,-5) (-1/2,3/4) and (-5/3,1/3)

16 Using the point-slope form to solve problems Whammo Toys can sell 2000 Frisbees a $6. If they raise the price to $8, sales will fall to 1500 a day. Are these two points that can be used to write an equation in point-slope form? The two points are (6,2000) and (8,1500) Then m = ( )/(8-6) = -500/2 = -250 Using the first point (6,2000), the point-slope form is: y 2000 = -250(x 6) Distributing and adding 2000 to each side results in: y = -250x , which is in slope-intercept form

17 Whammo Toys (cont) The Frisbee equation in slope-intercept form is y = -250x What would be the sales if price was set at $7.50 Since x = price, substitute 7.5 in for x to find the sales (y) y = -250(7.5) y = 1625 Whammo would sell 1625 Frisbees a $ Frisbee sales by price 7.5,

18 Solve the following using point-slope form A bottle holds some water, and is under a faucet. After 3 seconds of filling, it holds 10 ounces of water. After 20 seconds, it holds 24 ounces of water. Let y be the amount of water and x be the number of seconds of filling. Write an equation that shows the amount of water in the bottle in terms of x. Hint: in terms of x means solve for y

19 Chapter 3 Review Page

20 Section 3.1 Rectangular Coordinate System The rectangular coordinate system has an x-axis (horizontal) and the y-axis (vertical) The point of intersection (0,0) is called the origin An ordered pair is a point on the graph with an x and y coordinate: (x,y) An ordered pair is a solution of an equation with 2 variables x & y There are many ordered pairs as the solution of equations of 2 variables

21 Rectangular Coordinate System 3.1 An ordered pair point exists in one of four Quadrants I, II, III or IV Points with positive x and y are in Quadrant I Points with positive x and negative y are in Quadrant IV Negative x and negative y are in Quadrant III Negative x and positive y are in Quadrant II

22 More on Section 3.1 Complete the ordered pair (0,y) for the equation 2x 5y = -10 Substitute 0 for x Then -5y = -10 Therefore: y = 2 Which quadrant do the following points lie in: (3,5) (-2,-2) (4,-6)

23 Section 3.2 Graphing linear equations A linear equation in two variables is an equation that can be written in the Standard form Ax + By = C To graph a linear equation in 2 variables, find three ordered pair solutions. Plot the three points on the graph and draw the connecting line. Graph 2x 5y = 7 Set x = 0 Then set y = 0 Then produce one more point x 0-7/5 y 7/

24 Section Intercepts An intercept of a graph is the point where the line intersects an axis The x-intercept is the point where y = 0 The y-intercept is the point where x = 0 To find the intercepts of an equation: x-intercept: let y=0 and solve for x y-intercept: let x=0 and solve for y , -2 3,

25 Section 3.3 Intercepts (cont) The graph of x = c is a vertical line with x intercept (c,0) There is no y-intercept for a vertical line The graph of y = c is a horizontal line with y- intercept (0,c) There is no x-intercept for a horizontal line

26 Section 3.4 Slope and rate of change The slope of a line through points (x 1,y 1 ) and (x 2,y 2 ) is designated as m and is equal to: m = (y 2 y 1 ) (x 2 x 1 ) Slope is the amount of vertical change over the amount of horizontal change Also know as rise over run The slope of a line through points (-1,6) and (-5,8) is: m = (8-6)/(-5 (-1)) = (8-6)/(-5 + 1) = 2/-4 = -1/2

27 Section Slope If given an equation in standard form: -3x + y = 5 Step 1: find the x-intercept (set y = 0): -3x + 0 = 5; x = -5/3; x-intercept: (-5/3,0) Step 2: find the y-intercept (set x = 0): -3(0) + y = 5; y = 5; y-intercept: (0,5) Step 3: find m = (5-0)/(0-(-5/3)) m = 5 5/3 = 5 * 3/5 = 3

28 Section 3.4 Slope (cont) The slope of horizontal lines (ex: y = -3) have zero slope 6 5 The slope of vertical lines (ex: x = 2) have undefined slope The slopes of parallel lines are identical m = 1 m =

29 Section 3.4 Slope (cont.) The slopes of 2 perpendicular lines are negative reciprocals of one another m 1 * m 2 = -1 Line 1 has slope of 2; Line 2 has slope of -2; are they perpendicular? m 1 * m 2 = 2 * -2 = -4 Line 1 and line 2 are NOT perpendicular

30 Section 3.5 Equations of lines The Standard form of an equation of 2 variables: Ax + By = C; where A, B, & C are real numbers The Slope-intercept form of an equation of 2 variables: y = mx + b; where m is the slope of the line and b is the y-intercept point (0,b) The Point-slope form of an equation of 2 variables: y y 1 = m(x x 1 ); where m is the slope and (x 1,y 1 ) is a point on the line

31 Section 3.5 Equations of lines (cont) Find the equation of a line with slope 1/8 and y-intercept (0,12) Set m= 1/8 and b = 12 Therefore: y = 1/8x + 12 in Slope-intercept form Subtract 1/8x from each side -1/8x + y = 12 is the equation in Standard form

32 Section 3.5 cont. What is the equation of the line that has slope = -1/4 and passes through (2,2)? Since (2,2) is not an intercept, we have to use Point-slope form y 2 = -1/4(x 2) is the equation in point-slope form. Now solve for y by distributing and adding 2 to both sides y = -1/4x + ½ + 2 = -1/4x + 5/2 y = -1/4x + 5/2 is the equation in slope-intercept form. Finally, add 1/4x to each side 1/4x + y = 5/2 is the equation in standard form

33 Finally, some story problems A parallelogram is 42 meters in perimeter. One side is x and the other is 2y. Write the equation of 2 variables. The perimeter 42 = 2(x) + 2(2y) = 2x + 4y 2x + 4y = 42 is the equation in standard form. Now, solve for y to get the equation in slope-intercept form 4y = -2x + 42; y = -1/2x ; m = -1/2 Negative m means x gets smaller as y gets bigger Now, what is x when y is 8? 8 = -1/2x ; -1/2x = -2.5; x = 5

34 Section 3 movie ticket sales example In 2000, 142 million movie tickets were sold. In 2003, 157 million tickets were sold. Find the slope of the line for this equation, then write the slope as a rate of change. Write two ordered pairs (2000,142) and (2003,157) Find the slope: ( )/( ) = 15/3 = 5 What does this mean? Since x is the number of years, and y is millions of tickets sold: Each year, 5 more million tickets are sold.

35 Last minute quiz The equation y y 1 = m(x x 1 ) is a linear equation in what kind of form? A) Standard form B) Point-Slope form C) Polynomial form D) Slope-intercept form Write A, B, C, or D on a blank piece of paper, don t sign it, and hand it to me on the way out

E. Slope-Intercept Form and Direct Variation (pp )

E. Slope-Intercept Form and Direct Variation (pp ) and Direct Variation (pp. 32 35) For any two points, there is one and only one line that contains both points. This fact can help you graph a linear equation. Many times, it will be convenient to use the