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General Mathematics ADE 101 Unit LECTURE No 14 TYPES OF LINEAR EQUATIONS Today’s Objectives Knowledge Test Point – Slope Form To write an equation of a line in point – slope form, all you need is … (x1, y1) … Any Point On The Line … … The Slope m … Once you have these two things, you can write the equation as y – y1 = m (x – x1) That’s “y minus the yvalue of the point equals the slope times the quantity of x minus the xvalue of the point” Example Write the equation of the line that goes through the point (2, –3) and has a slope of 4 Point = (2, –3) Slope = 4 Starting with the point – slope form y – y1 = m (x – x1) Plug in the yvalue, the slope, and the xvalue to get y + 3 = 4 (x – 2) Notice, that when you subtracted the “–3” it became “+3” Example Write the equation of the line that goes through the point (–4, 6) and has a slope of Point = (–4, 6) Slope = Starting with the point – slope form y – y1 = m (x – x1) Plug in the yvalue, the slope, and the xvalue to get y–6= (x + 4) Notice, that when you subtracted the “–4” it became “+4” Example Write the equation of the line that goes through the points (6, –4) and (2, 8) We have two points, but we’re missing the slope. Using the formula for slope, we can find the slope to be y2 – y1 8- ( - 4) 12 m= = =- x2 – x1 2- -4 To use point – slope form, we need a point and a slope. Since we have two points, just pick one … IT DOESN’T MATTER … BOTH answers are acceptable… more on why later Using the first point, we have, Using the second point, we have, Point = (6, –4) Point = (2, 8) Slope = –3 Slope = –3 y + 4 = –3 (x – 6) y – 8 = –3 (x – 2) Slope-intercept Form v An equation whose graph is a straight line is a linear equation. Since a function rule is an equation, a function can also be linear v m = slope v b = yintercept Y = mx + b (if you know the slope and where the line crosses the yaxis, use this form) Writing Equations in Slope – Earlier we wrote an equation of the line that went through the points (6, –4) and (2, Intercept Form 8) . Sometimes, we want the line written in a different form To change a pointslope equation in slopeintercept form, solve for y and simplify the right side of the equation. Solve for y: Add or subtract the yvalue of the point to both sides Simplify: Distribute the slope and then combine like terms Here are the two answers we had from the earlier example y + 4 = –3 (x – 6) y – 8 = –3 (x – 2) Subtract 4 from both sides Add 8 to both sides y = –3 (x – 6) – 4 y = –3 (x – 2) + 8 Distribute –3 and combine like terms y = –3x + 18 – 4 y = –3x + 14 SIMPLIFY Distribute –3 and combine like terms y = –3x + 6 + 8 y = –3x + 14 Notice … They’re the same! Example Write the equation of the line in slope-intercept form that goes through the point (6, 2) and has slope3 y–2= Begin in point-slope form: (x – 6) Add 2 to both sides Solve for y: y = (x – 6) + 2 Distribute: y= Combine Like Terms: y= x–2 x–4+2 Write the equation of a line in slope-intercept form that passes through points (3, -4) and (-1, 4) 1) Find the slope 4 – (4) 8 1 – 3 4 m = 2 2) Choose either point and substitute. Solve for b y = mx + b(3, 4) 4 = (2)(3) + b 4 = 6 + b 2 = b Substitute m and b in equation Y = mx + b Y = 2x + 2 Write the equation of the line in slope-intercept form that passes through each pair of points 1) (1, 6) and (2, 6) 2) (0, 5) and (3, 1) 3) (3, 5) and (6, 6) 4) (0, 7) and (4, 25) 5) (1, 1) and (3, 3) The slope is 3, use the slope to plot the second point Graphing an Equation y = 3x -1 The yintercept is 1, so plot point (0, 1) Draw a line through the two points Point-Slope Form Writing an equation when you know a point (2, 5) and the slope m = 2 Other Forms of Linear So far, we have discussed only pointslope form. There are other forms of equations Equations that you should be able to identify as a line and graph if necessary. Horizontal Line: y = c , where c is a constant Example: y = 3 Vertical Line: x = c , where c is a constant Example: x = –6 Slope – Intercept Form: y = mx + b m = the slope of the line … b = the yintercept Example: y = 3x – 6 Standard Form: Ax + By = C A, B, and C are integers Example: 3x + 4y = –36 To write equations in the last two forms, start in point – slope form and rearrange the variables to match the correct format The next few slides will cover how to do this Writing Equations In Standard The last form of a linear equation we are going to cover is called Standard Form Form Ax + By = C , where A, B, and C are integers If you needed to write an equation of a line in standard form, you would start in pointslope form or slopeintercept form, depending on what information you are given In both cases, you must put all variables on the left side and all constant values on the right side If any of the coefficients (A, B, or C) are NOT integers, then you must eliminate any fractions or decimals by multiplying every term in the equation by the appropriate factor Let’s a couple more to make sure you are expert at this Given m = 2/3, b = 12, Write the equation of a line in slopeintercept form Y = mx + b Y = 2/3x – 12 One last example… Given m = 5, b = 1 Write the equation of a line in slopeintercept form Y = mx + b Write an equation of each line Use points (0, 1) and (2, 0) Use points (0, 1) and (3, 1) Given the slope and y-intercept, write the equation of a line in slope-intercept form 1) m = 3, b = 14 2) m = ½, b = 4 3) m = 3, b = 7 4) m = 1/2 , b = 0 5) m = 2, b = 4 Using slope-intercept form to find slopes and y-intercepts The graph at the right shows the equation of a line both in standard form and slope-intercept form You must rewrite the equation 6x – 3y = 12 in slope-intercept to be able to identify the slope and yintercept In the graph below, use the information provided to write the equation of the line Use what you know about writing an equation in slope-intercept form Slope = 2 and point (2,7) Do you think you can use the same method to find the yintercept in the graph below? Here we must use a different form of writing an equation and that form is called point-slope Slope = 7/3 and point (2,7) y – y = m ( Suppose you know that a line passes through the point (3, 4) with slope 2. You can quickly write an equation of the line using the x and y coordinates of the point and using the slope The pointslope form of the equation of a nonvertical line that passes through the (x1, y1) with slope m Point-Slope Form and Writing Equations Let’s try a couple Using pointslope form, write the equation of a line that passes through (4, 1) with slope 2 y – y1 = m(x – x1) y – 1 = 2(x – 4)Substitute 4 for x1, 1 for y1 and 2 for m Write in slopeintercept form y – 1 = 2x + 8 y = 2x + 9 One last example Using pointslope form, write the equation of a line that passes through (1, 3) with slope 7 y – y1 = m(x – x1) y – 3 = 7[x – (1)] y – 3 = 7(x + 1) Write in slopeintercept form y – 3 = 7x + 7 y = 7x + 10