3-6. Lines in the Coordinate Plane Going Deeper E X A M P L E. Slope-Intercept Form. Point-Slope Form. Writing Equations of Parallel Lines

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Name Class Date 3-6 Lines in the Coordinate Plane Going Deeper Essential question: How can you use slope to write equations of lines that are parallel or perpendicular?

You can write a linear equation in different forms depending upon the information you are given and the problem you are trying to solve.

.5 E X A M P L E Writing Equations of Parallel Lines Write the equation of each line in slope-intercept form.

1 Name Class Date 3-6 Lines in the Coordinate Plane Going Deeper Essential question: How can you use slope to write equations of lines that are parallel or perpendicular? Recall that a linear function can be expressed as a linear equation. You can write a linear equation in different forms depending upon the information you are given and the problem you are trying to solve. Slope-Intercept Form The equation of a line with slope m and y-intercept b is y = mx + b. Point-Slope Form The equation of a line with slope m that passes through the point ( x 1, y 1 is. 1 G-GPE..5 E X A M P L E Writing Equations of Parallel Lines Write the equation of each line in slope-intercept form. A The line parallel to y = -x + 3 that passes through (1, -) The given line is in slope-intercept form and its slope is. The required line has slope because parallel lines have the same slope. y - = (x - ) Substitute for m, x 1, and y 1. y + = Simplify each side of the equation. y = Write the equation in slope-intercept form. B The line that passes through (, 3) and is parallel to the line through (1, -) and (7, 1) The slope of the line through (1, -) and (7, 1) is m = y - y 1 x - x 1 = - - = =. So, the required line has slope. y - = (x - ) Substitute for m, x 1, and y 1. y = Simplify and write slope-intercept form. Chapter Lesson 6

2 REFLECT 1a. In Part A, how can you check that you wrote the correct equation? 1b. In Part A, once you know the slope of the required line, how can you finish solving the problem using the slope-intercept form of a linear equation? G-GPE..5 E X A M P L E Writing Equations of Perpendicular Lines Write the equation of the line perpendicular to y = 3x - 8 that passes through (3, 1). Write the equation in slope-intercept form. A First find the slope of the required line. The given line is in slope-intercept form and its slope is. Let the required line have slope m. Since the lines are perpendicular, the product of their slopes is -1. So, m = -1, and therefore, m =. B Now use point-slope form to find the equation of the required line. y - = (x - ) Substitute for m, x 1, and y 1. y - = Distributive Property y = Write the equation in slope-intercept form. REFLECT a. How do you find the slope of the given line? b. How can you use graphing to check your answer? Chapter Lesson 6

3 c. Confirm your answer by graphing on the grid below. y x p r a c t i c e Write the equation of each line in slope-intercept form. 1. The line with slope 3 that passes through (0, 6). The line with slope - that passes through (0, -5) 3. The line with slope -1 that passes through (3, 5). The line with slope 5 that passes through (, -5) 5. The line parallel to y = 5x + 1 that passes through (3, 8) 6. The line parallel to y = -3x - that passes through (-, 7) 7. The line that passes through (-1, 0) and is parallel to the line through (0, 1) and (, -3) 9. The line parallel to x - 3y = -1 that passes through (-3, ) 8. The line that passes through (3, 5) and is parallel to the line through (3, 3) and (-3, -1) 10. The line parallel to 3x + y = 8 that passes through (0, -) 11. Use the slope-intercept form of a linear equation to prove that if two lines are parallel then they have the same slope. (Hint: Use an indirect proof. Assume the lines have different slopes, m 1 and m. Write the equations of the lines and show that there must be a point of intersection.) Chapter Lesson 6

4 Write the equation of each line in slope-intercept form. 1. The line perpendicular to y = 1_ x + 1 that 13. The line perpendicular to y = -x + that passes through (1, ) passes through (-1, -7) 1. The line that passes through (1, ) and is perpendicular to the line through (3, -) and (-3, 0) 15. The line that passes through (-, 3) and is perpendicular to the line through (0, 1) and (-3, -1) 16. The line perpendicular to y = x + 5 that passes through (, 1) 17. The line perpendicular to 3x + y = 8 that passes through (0, -) 18. Error Analysis A student was asked to find the equation of the line perpendicular to y - x = 1 that passes through the point (, 3). The student s work is shown at right. Explain the error and give the correct equation. The given line has slope -, so the required line has slope 1. y - 3 = 1 (x - ) Substitute for m, x 1, y 1. y - 3 = 1 x - Distributive Property y = 1 x + 1 Add 3 to both sides. 19. Are the lines given by the equations -x + y = 5 and -x + y = 1 parallel, perpendicular, or neither? Why? 0. Consider the points A(-7, 10), B(1, 7), C(10, -), and D(-8, -3). Which two lines determined by these points are perpendicular? Explain. Chapter 3 10 Lesson 6

the line with x-intercept and y-intercept slope-intercept form 1 in slope-intercept form Graph each line. 5. 3 y + 3 = ( x + 1) 6.

5 Name Class Date Additional Practice 3-6 Write the equation of each line in the given form. 1. the horizontal line through (3, 7) in. the line with slope point-slope form point-slope form 8 through (1, 5) in the line through, and (, 1) in. the line with x-intercept and y-intercept slope-intercept form 1 in slope-intercept form Graph each line y + 3 = ( x + 1) 6. y = x + 3 Determine whether the lines are parallel, intersect, or coincide x 5y = 0, y + 1 = ( x + 5) 5 1 y + = x, x = 1+ y y = ( x 3), + y = x An aquifer is an underground storehouse of water. The water is in tiny crevices and pockets in the rock or sand, but because aquifers underlay large areas of land, the amount of water in an aquifer can be vast. Wells and springs draw water from aquifers. 10. Two relatively small aquifers are the Rush Springs (RS) aquifer and the Arbuckle- Simpson (AS) aquifer, both in Oklahoma. Suppose that starting on a certain day in 1985, 5 million gallons of water per day were taken from the RS aquifer, and 8 million gallons of water per day were taken from the AS aquifer. If the RS aquifer began with 500 million gallons of water and the AS aquifer began with 3000 million gallons of water and no rain fell, write a slope-intercept equation for each aquifer and find how many days passed until both aquifers held the same amount of water. (Round to the nearest day.) Chapter 3 11 Lesson 6

6 Problem Solving Use the following information for Exercises 1 and. Josh can order 1 color ink cartridge and black ink cartridges for his printer for $78. He can also order 1 color ink cartridge and 1 black ink cartridge for $ Let x equal the cost of a color ink. What is the cost of each cartridge? cartridge and y equal the cost of a black ink cartridge. Write a system of equations to represent this situation. 3. Ms. Williams is planning to buy T-shirts for the cheerleading camp that she is running. Both companies total costs would be the same after buying how many T-shirts? Use a graph to find your solution. Art Creation Fee Cost per T-shirt Company A $70 $10 Company B $50 $1 Choose the best answer.. Two floats begin a parade at different 5. A piano teacher charges $0 for each half times, but travel at the same speeds. hour lesson, plus an initial fee of $50. Which is a true statement about the Another teacher charges $0 per hour, lines that represent the distance traveled plus a fee of $50. Which is a true statement by each float at a given time? A The lines intersect. about the lines that represent the total cost by each piano teacher? B The lines are parallel. F The lines intersect. C The lines are the same. G The lines are parallel. D The lines have a negative slope. H The lines are the same. 6. Serina is trying to decide between two J The lines have a negative slope. similar packages for starting her own Web site. Which is a true statement? A Both packages cost $35.50 for 5 months. B Both packages cost $95 for 10 months. C Both packages cost $355 for 15 months. D The packages will never have the same cost. Design and Setup Monthly Fee to Host Package A $ $1.50 Package B $ $1.00 Chapter 3 1 Lesson 6

1. a pair of parallel segments. 2. a pair of skew segments. 3. a pair of perpendicular segments. 4. a pair of parallel planes

Practice Lines and Angles For Exercises 1 4, identify each of the following in the figure. 1. a pair of parallel segments 2. a pair of skew segments 3. a pair of perpendicular segments 4. a pair of parallel