คาบ 5 7

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คาบ 5 7

  1. 1. Digital Lesson<br />Linear Equations in Two Variables<br />
  2. 2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />2<br />y<br />This is the graph of the equation 2x + 3y = 12.<br />(0,4)<br />(6,0)<br />x<br />2<br />-2<br />Equations of the form ax + by = c are called linear equations in two variables.<br />The point (0,4) is the y-intercept.<br />The point (6,0) is the x-intercept.<br />Linear Equations<br />
  3. 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />3<br />y<br />1<br />1<br />m =<br />2<br />4<br />x<br />2<br />-2<br />m = -<br />The slope of a line is a number, m, which measures its steepness.<br />m is undefined<br />m = 2<br />m = 0<br />Slope of a Line<br />
  4. 4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />4<br />y2 – y1<br />, (x1 ≠ x2). <br />m =<br />x2 – x1<br />(x2, y2)<br />y<br />y2–y1<br />change in y<br />(x1, y1)<br />x2–x1<br />change in x<br />x<br />The slope of the line passing through the two points (x1, y1) and (x2, y2) is given by the formula <br />The slope is the change in y divided by the change in x as we move along the line from (x1, y1) to (x2, y2).<br />Slope Formula<br />
  5. 5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />5<br />y2 – y1<br />5 – 3<br />2<br />1<br />m =<br />= <br />= <br />= <br />x2 – x1<br />2<br />4 – 2<br />y<br />x<br />Example: Find the slope of the line passing through the points (2,3) and (4,5). <br />Use the slope formula with x1= 2, y1 = 3, x2 = 4, and y2 = 5.<br />(4, 5)<br />2<br />(2, 3)<br />2<br />Example: Find Slope<br />
  6. 6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />6<br />A linear equation written in the form y = mx + b is in slope-intercept form.<br />The slope is m and the y-intercept is (0, b).<br />To graph an equation in slope-intercept form:<br />1. Write the equation in the form y = mx + b. Identify m and b.<br />2. Plot the y-intercept (0,b).<br />3. Starting at the y-intercept, find another point on the line using the slope.<br />4. Draw the line through (0, b) and the point located using the <br /> slope.<br />Slope-Intercept Form<br />
  7. 7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />7<br />y<br />x<br />change in y<br />2<br />= <br />m =<br />1<br />change in x<br />(0,-4)<br />(1, -2)<br />Example: Graph the line y = 2x– 4.<br />The equation y = 2x– 4 is in the slope-intercept form. So, m = 2 and b = -4.<br />2. Plot the y-intercept, (0,-4).<br />3. The slope is 2.<br />2<br />4. Start at the point (0,4). Count 1 unit to the right and 2 units up to locate a second point on the line.<br />1<br />The point (1,-2) is also on the line.<br />5. Draw the line through (0,4) and (1,-2). <br />Example: y=mx+b<br />
  8. 8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />8<br />1<br />1<br />1<br />2<br />2<br />2<br />Example:The graph of the equation y – 3 = -(x – 4) is a line of slope m = - passing through the point (4,3).<br />y<br />m = -<br />8<br />(4, 3)<br />4<br />x<br />4<br />8<br />A linear equation written in the form y–y1 = m(x – x1) is in point-slope form. <br />The graph of this equation is a line with slope m passing through the point (x1, y1).<br />Point-Slope Form<br />
  9. 9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />9<br />Point-slope form<br />y – y1 = m(x – x1)<br />y – y1 = 3(x – x1)<br />Let m = 3.<br />y – 5 = 3(x – (-2))<br />Let (x1, y1)= (-2,5).<br />y – 5 = 3(x + 2)<br />Simplify.<br />y = 3x + 11<br />Slope-intercept form<br />Example: Write the slope-intercept form for the equation of the line through the point (-2,5) with a slope of 3. <br />Use the point-slope form, y – y1 = m(x – x1), with m = 3 and (x1, y1)= (-2,5).<br />Example: Point-Slope Form<br />
  10. 10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />10<br />m =<br />Calculate the slope.<br />1<br />1<br />3<br />3<br />Point-slope form<br />y – y1 = m(x – x1)<br />2<br />5 – 3<br />1<br />= - <br />= - <br />Use m = -and the point (4,3).<br />-2 – 4<br /> 6<br /> 3<br />y – 3 = - (x – 4)<br />13<br />Slope-intercept form<br />3<br />y = - x +<br />Example: Write the slope-intercept form for the equation of the line through the points (4,3) and (-2,5).<br />Example: Slope-Intercept Form<br />
  11. 11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />11<br />y<br />(0, 4)<br />y = 2x + 4<br />x<br />y = 2x– 3<br />(0, -3)<br />Two lines are parallel if they have the same slope.<br />If the lines have slopes m1 and m2, then the lines are parallel whenever m1 = m2.<br />Example: The lines y = 2x – 3 and y = 2x + 4 have slopes m1 = 2 and m2 = 2.<br />The lines are parallel.<br />Example: Parallel Lines<br />
  12. 12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.<br />12<br />y<br />1<br />1<br />1<br />1<br />m2= -<br />3<br />3<br />3<br />m1<br />x<br />Example: The lines y = 3x – 1 and y = - x + 4 have slopes m1 = 3 and m2 = - .<br />y = - x + 4<br />Two lines are perpendicular if their slopes are negative reciprocals of each other.<br />If two lines have slopes m1 and m2, then the lines are perpendicular whenever<br />or m1m2 = -1.<br />y = 3x– 1<br />(0, 4)<br />(0, -1)<br />The lines are perpendicular.<br />Example: Perpendicular Lines<br />

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