Linear Equations in Two Variables Digital Lesson
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 A linear equation written in the form y = mx + b is in slope-intercept form. To graph an equation in slope-intercept form: 1. Write the equation in the form y = mx + b. Identify m and b. The slope is m and the y-intercept is (0, b). 2. Plot the y-intercept (0, b). 3. Starting at the y-intercept, find another point on the line using the slope. 4. Draw the line through (0, b) and the point located using the slope. Slope-Intercept Form
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Finding the Slope of a Line (straight line)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Definition of the Slope of a Line
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 1 Example: Graph the line y = 2x – Plot the y-intercept, (0, - 4). 1.The equation y = 2x – 4 is in the slope-intercept form. So, m = 2 and b = The slope is 2. The point (1, -2) is also on the line. 1 = change in y change in x m = 2 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. 2 x y 5. Draw the line through (0, 4) and (1, -2). Example: y=mx+b (0, - 4) (1, -2)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example 1. Graph the line with slope 3/2 that pass though the point (-2, -2) 2. Consider, lines that are parallel to the axes. - Parallel to the x-axis - Parallel to the y-axis - Ratio (x-axis and y-axis have the same unit of measure) - Rate or rage of change (x-axis and y-axis have the different unit of measure)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 A linear equation written in the form y – y 1 = m(x – x 1 ) is in point-slope form. The graph of this equation is a line with slope m passing through the point (x 1, y 1 ). Example: The graph of the equation y – 3 = - (x – 4) is a line of slope m = - passing through the point (4, 3) (4, 3) m = x y Point-Slope Form
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Write the slope-intercept form for the equation of the line through the point (-2, 5) with a slope of 3. Use the point-slope form, y – y 1 = m(x – x 1 ), with m = 3 and (x 1, y 1 ) = (-2, 5). y – y 1 = m(x – x 1 ) Point-slope form y – y 1 = 3(x – x 1 ) Let m = 3. y – 5 = 3(x – (-2)) Let (x 1, y 1 ) = (-2, 5). y – 5 = 3(x + 2) Simplify. y = 3x + 11 Slope-intercept form Example: Point-Slope Form
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: Write the slope-intercept form for the equation of the line through the points (4, 3) and (-2, 5). y – y 1 = m(x – x 1 ) Point-slope form Slope-intercept form y = - x – 3 -2 – 4 = Calculate the slope. m = Example: Slope-Intercept Form Use m = - and the point (4, 3). y – 3 = - (x – 4) 1 3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Prediction
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Summary of Equations of Lines
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Parallel and Perpendicular Lines 1.Two distinct nonvertical lines are parallel if and only if their slopes are equal. That is, 2. Two nonvertical lines are perpendicular if and only if their slopes are negative reciprocals of each other. That is,
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Two lines are parallel if they have the same slope. If the lines have slopes m 1 and m 2, then the lines are parallel whenever m 1 = m 2. Example: The lines y = 2x – 3 and y = 2x + 4 have slopes m 1 = 2 and m 2 = 2. The lines are parallel. x y y = 2x + 4 (0, 4) y = 2x – 3 (0, -3) Example: Parallel Lines
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Example Find the slope-intercept forms of the equations of the lines that pass through the point (2,-1) and are (a) parallel to and (b) perpendicular to the line 2x – 3y = 5