Writing Linear Equations: Mastering the Slope-Intercept Form

Math A100 IUSB Sec. 3.5 detailed notes – Week 6 Writing the y = mx + b equation The Basic Linear Equation is y = mx + b. It is called the slope-intercept form. I will NOT ask you to define it on a quiz or test, but you DO have to recognize the wording, because the directions for the HW/quizzes/test asks for the answer in slope-intercept form, the answer must be in y = mx + b form. In order to write the equation of the line, you need two things: 1) The slope, “ m ” 2) The y value of the y-intercept, “ b ” - keep in mind that the y-intercept is an ordered pair where the x value = 0, i.e. (0, b) NOTE : “ y ” , “ = ” , “ x ” , and “ + or –“ are all standard to the equations, i.e. they need to be there. In other words, you only have to fill in values for “ m ” and “ b ” (i.e. y = ____x + _____) There are several different combinations of things they can give to you and ask you to write the equations of the line: 1) The line on a grid – look at the graph to find the following two things a. the point where the line crosses the y-axis. The y value of that is “ b ” . Example 1: the line crosses the y axis at (0, -5)  “ b ” = -5 Example 2: the line crosses the y axis at (0, 7)  “ b ” = 7 Example 3: the line crosses through the origin  “ b ” = 0, and does not have to be included in the equation b. pick two points the are easy to see and calculate the slope (from sec. 3.6) Once you have both “ b ” and “ m ” , plug them into the basic equations to get the equation of the line Example: The line crosses the y-axis at (0, -2), “ b ” = -2 Pick two points: (-6, 0) & (3, -3) The slope is down 3 and to the right 9, so “ m ” = -3/9 = -1/3 Plug them into the basic equation: y = -1/3x - 2 Reminder: in horizontal lines, the slop = 0, so the 0*x part falls out and your equation is just “ y = ____ ” In vertical lines, the slope is Undefined, so your equation is just “ x = _____ ” 2) Slope and the y-intercept – if given a slope and the y-intercept, plug “ m ” and “ b ” directly into the basics equation Example: m = 4/5 and y-intercept is (0, 6). The equation of the line is y = 4/5x + 6

3) The slope and a point which is NOT the y-intercept (i.e. does NOT have “ b ” in it) You have three options to choose from: Option A : Graph the line on a grid and get the answer from there (see #1 above) Option B : Use y = mx + b to find “ b ” – plug in the m, x, and y values and solve for “ b ” Example 1: m = 3/2 and thru the point (-6, 5)  plug in 3/2 for m, -6 for x, & 5 for y 5 = (3/2)(-6) + b do the multiplication to get: 5 = -9 + b add 9 from both sides to get: 14 = b Now plug “ m ” and “ b ” into the basic equation: y = mx + b becomes y = 3/2x + 14 Example 2: m = -4 and thru the point (-1, 5)  plug in 3/2 for m, -6 for x, & 5 for y 5 = (-4)(-1) + b do the multiplication to get: 5 = 4 + b subtract 4 from both sides to get: 1 = b Now plug “ m ” and “ b ” into the basic equation: y = mx + b becomes y = -4x + 1 Option C : Use the point-slope formula : y – y 1 = m(x – x 1 )  plug in the values for m, x 1, and y 1 then put it back into y = mx + b form Example 1: m = 3/2 and thru the point (-6, 5)  plug in 3/2 for m, -6 for x 1 , & 5 for y 1 y – 5 = (3/2)(x - -6) Clean it up to get: y – 5 = (3/2)(x + 6) distribute to get: y – 5 = 3/2x + 9 add 5 from both sides to get: y = 3/2x + 14 Example 2: m = -4 and thru the point (-1, 5)  plug in 3/2 for m, -6 for x, & 5 for y y – 5 = (-4)(x - -1) Clean it up to get: y – 5 = (-4)(x + 1) distribute to get: y – 5 = -4x - 4 add 5 from both sides to get: y = -4x + 1 4) Two points, where one IS the y-intercept – find the slope using the points (see #1 above), then do like in #2 above. Example 1: Write the equation of the line that goes through (-6, 5) and (0, -3) Find the slope m = -8/6 = -4/3 y-intercept is (0, -3), so “ b ” = -3. The equation of the line is y = -4/3x - 3

5) Two points, in which neither one is the y-intercept - find the slope using the points (see #1 above), then do like in #3 above. NOTE: you can use either point (both work equally as well!!!) Example 1: Write the equation of the line that goes through (-6, 5) and (4, 10) Find the slope m = 5/10 = ½ Let ’ s use (4, 10) 10 = (1/2)(4) + b do the multiplication to get: 10 = 2 + b subtract 2 from both sides to get: 8 = b Now plug “ m ” and “ b ” into the basic equation: y = mx + b becomes y = 1/2x + 8 OR m = 1/2 and let ’ s use the point (-6, 5)  plug in 1/2 for m, -6 for x 1 , & 5 for y 1 y – 5 = (1/2)(x - -6) Clean it up to get: y – 5 = (1/2)(x + 6) distribute to get: y – 5 = 1/2x + 3 add 5 from both sides to get: y = 1/2x + 8 Special cases Horizontal lines Recall from previously, the equation of a horizontal line is y = ____. In horizontal lines, every point has the same y-coordinate, i.e. it really does not matter what the x-coordinate is. So, to write the equations of the line, determine what the y-coordinate of all the points on the line is and plug that value into the equation above. Example 1: Write the equation of the horizontal line going thru (5, -6) The y-coordinate = - 6, so every point on this line has a y-coordinate of - 6  y = - 6  equation Vertical lines Recall from previously, the equation of a vertical line is x = ____. In vertical lines, every point has the same x-coordinate, i.e. it really does not matter what the y-coordinate is. So, to write the equations of the line, determine what the x-coordinate of all the points on the line is and plug that value into the equation above. Example 1: Write the equation of the vertical line going thru (5, -6)

The x-coordinate = , so every point on this line has a x-coordinate of 5  x = 5  equation Videos that may help: The Slope-Intercept form of the Equation of a Line http://www.youtube.com/watch?v=eDf9Kxh3XAA http://www.youtube.com/watch?v=IkHJgg-Xjdk Equations of Horizontal and Vertical Lines http://www.youtube.com/watch?v=iSgftOjXIfw