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5.2 Writing Linear Equations Given the Slope and a Point Today we will learn how to: ◦ Use slope and any point on a line to write an equation of the line ◦ Use a linear model to make predictions about a real-life situation We already know how to write an equation given the slope and the y-intercept. Now we will learn how to write an equation given its slope and any point on the line. 1. First find the y-intercept. Substitute the slope m and the coordinates of the given point (x, y) into the slope-intercept form, y = mx + b. Then solve for the yintercept b. 2. Then write an equation of the line. Substitute the slope m and the y-intercept b into the slope-intercept form y = mx + b. Example 1 Write an equation of the line that passes through the point (-3, 0) and has a slope of 1 3 1. Substitute the slope m and the coordinates into slope-intercept form: 1 0= 3 (-3) + b 0=-1+ b 1=b 2. Substitute slope m and the y-intercept, b into the equation: 1 m= 3 , b= 1 y= 1 x + 1 3 Example 2 Write an equation of the line that is parallel to the line y = -3x – 2 through the point (3, -4). Find the slope of the line that is parallel to the line y=-3x-2. So the slope of the line is -3 (when the lines are parallel they have the same slope) 1. Find the y-intercept Substitute the slope and the coordinates of the given point into the equation y=mx+b -4=-3(3) + b -4=-9 + b 5=b 2. Write an equation of the line m=-3, b=5 y=-3x + 5 Writing and Using a Linear Model VACATION TRIPS Between 1985 and 1995, the number of vacation trips in the United States taken by United States residents increased by about 26 million per year. In 1993, United States residents went on 740 million vacation trips within the United States. a. Write a linear equation that models the number of vacation trips y (in millions) in terms of the year t. Let t be the number of years since 1985. b. Estimate the number of vacation trips in the year 2005 SOLUTION a. The number of trips increased by about 26 million per year, so you know the slope is m = 26. You also know that (t, y) = (8, 740) is a point on the line, because 740 million trips were taken in 1993, 8 years after 1985. y = mt + b Write slope-intercept form. 740 = (26)(8) + b Substitute 26 for m, 8 for t, and 740 for y. 740 = 208 + b Simplify. 532 = b The y-intercept is b = 532. Write an equation of the line using m = 26 and b = 532. y = mt + b Write slope-intercept form. y = 26t + 532 Substitute 26 for m and 532 forb. Solution b. You can estimate the number of vacation trips in the year 2005 by substituting t = 20 into the linear model. y = 26t + 532 Write linear model. = 26(20) + 532 Substitute 20 for t. = 1052 Simplify. You can estimate that United States residents will take about 1052 million vacation trips in the year 2005. HOMEWORK Ch 5.1 w/s 5.3 – Writing Linear Equations Given Two Points Today we will learn how to: ◦ Write an equation of a line given two points on the line So far in this chapter, we were always given the slope Now, we will have to first find the slope Use the formula from Chapter 4 rise y2 y1 m run x2 x1 Find the slope. Substitute the coordinates of the two given points into the formula for slope, 1. y2 y1 m x2 x1 2. 3. Find the y-intercept. Substitute the slope m and the coordinates of one of the points into the slope-intercept form, y mx b , and solve for the y-intercept, b. Write an equation of the line. Substitute the slope m and the y-intercept b into the slopeintercept form, y mx b Write an equation of the line that passes through the points (1,6) and (3,-4). 1. 2. 3. Find the slope Find the y-intercept Write the equation of the line 1. 2. 3. Find the slope of the line. m= -5 Find the y-intercept b=11 Write an equation of the line y=-5x+11 Lines in the same plane that do not intersect are called parallel lines. Parallel lines have the same slope. Lines that intersect at right angles are called perpendicular lines. The slopes of these lines are opposite reciprocals. (Or they multiply to equal -1) Opposite reciprocals???? 3….. 1 ... 4 y = 2x + 2 y = 4x - 2 neither 2x + 6y = 1 4x + 12y =3 parallel 1 x2 5 y 5 x 1 y perpendicular 1 y x4 2 Ch. 5.3 (Pg. 288-289) #18-46 EVEN