Unit 3: Writing Equations Chapter Review

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1 Unit 3: Writing Equations Chapter Review Part 1: Writing Equations in Slope Intercept Form. (Lesson 1) 1. Write an equation that represents the line on the graph. 2. Write an equation that has a slope of -3 and a y- intercept of Write an equation that has a slope of ½ and passes through the origin. 4. A day care center charges $9 per hour plus a $3 snack fee for a Parents Night Out Special. Write an equation that can be used to determine the price of sending a child to the day care center for x number of hours. Suppose a parent decides to send their child to the day care center for 5 hours. How much will this cost? Justify your answer. Part 2: Writing Equations in Standard Form (Lesson 2) 5. Rewrite the following equations in standard form. a. y = -4x 9 b. y = 6x + 2

2 6. Which term in the equation does not have an integer coefficient? 1/2x + 3y = 9 7. Write this equation in correct standard form: 1/2x + 3y = 9 Unit 3: Writing Equations 8. Write the following equation in standard form: 1/3x + 1/2y = 6 9. Write an equation, in standard form, for the line shown on the graph. a. b. Part 3: Standard Form Word Problems (Lesson 3) 10. A new 3-D movie is out and the theatre is charging $10.50 for adults and $7 for children. You have $50 to spend on the movies. Write an equation in standard form that represents the number of adult, a and child, c tickets that you can purchase. Suppose you are taking 3 adults, at most, how many children can you take to the movies? Justify your answer mathematically.

3 Part 4: Writing Equations Given Slope and a Point (Lesson 4) 11. Write an equation for the line that has a slope of -3 and passes through the point (5,-7). 12. A local tax service charges $65.25 an hour plus a filing fee. A three hour session costs $ Write an equation that can be used to find the total cost for any session. Part 5: Writing Equations Given Two Points (Lesson 5) 13. Write an equation for the line that passes through the points: (6,-2) (-4, 8) 14. In 1991,the cost of tuition for a private school was $5,000 per year. In 2012, the cost of the same private school is $12,500. Let x = 0 represent the year Write an equation that could be used to predict the cost of tuition for any given year. Predict what the tuition will be for the year In 2003, the cost of season football tickets was $2,200. In 2012, the cost of the same season tickets is $4,550. Let x = 0 represent the year Write an equation that could be used to predict the cost of the tickets for any given year. Predict what the cost of the tickets will be in the year Part 6: Writing Linear Equations in Point-Slope Form (Lesson 6) 16. Write an equation that has a slope of -3 and passes through the point (2,8) 17. Write an equation that passes through the points (1,9) and (3, -8). Explain the steps that you used to solve this problem. 18. Write an equation that has a slope of 1/2 and an x-intercept of 1. Explain how you solved this problem. 19. Write an equation that has a y-intercept of 22 and passes through the point (3, -7). 20. Write an equation that has an x-intercept of -5 and a y-intercept of 8.

4 21. Mei earned $38 on her investment in 5 months. She earned $62 on the same investment in 8 months. Write an equation that can be used to find the amount earned (y) on Mei s invest in x number of months. About how much will Mei earn after 15 months? About how much is Mei earning per month? Explain how you determined your answer. Part 7: Parallel and Perpendicular Lines (Lesson 7) Parallel Lines: Perpendicular lines: 22. Write an equation for a line that passes through (2,-8) and is perpendicular to a line whose slope is 5. Explain how you determined your answer. 23. Write an equation for a line that passes through (-3,6) and is parallel to the graph of x = Write an equation for a line that is perpendicular to the graph of 2x 2y = 10 and intersects the graph at its x-intercept.

5 Part 8: Scatter Plots and Line of Best Fit (Lesson 8) 25. The following data represents the total U.S. E-commerce sales from (Statistic from statista.com) Let x =0 represent the year Year E-commerce Sales (in billions of dollars) Using your graphing calculator, write an equation for the line of best fit for this data. What does the slope and y-intercept represent in the context of this problem.? Predict the amount of E-commerce sales in the U.S. for the year Explain how you determined your answer.

6 Unit 3: Writing Equations Chapter Review Answer Key Part 1: Writing Equations in Slope Intercept Form. (Lesson 1) 1. Write an equation that represents the line on the graph. In order to write this equation, we must identify the y-intercept and slope from the graph. The line crosses the y-axis at y = -6, so this is the y- intercept. From this point, to the next identifiable point on the graph, we count up1 unit and 2 units to the left, so the slope is -1/2. Y = mx+ b m = -1/2 b = -6 Y = -1/2x - 6 In order to write this equation, we must identify the y-intercept and slope from the graph. The line crosses the y-axis at y = 2, so this is the y- intercept. From this point, to the next identifiable point on the graph, we count up 3 units and 1 unit to the right, so the slope is 3. Y = mx+ b m = 3 b= 2 Y = 3x Write an equation that has a slope of -3 and a y- intercept of 10. Y = mx + b is the formula for writing an equation in slope intercept form. We need to know the slope (m) and the y-intercept (b). Slope (m) = -3 Y-intercept (b) = 10 Y = mx+b Y = -3x+ 10

7 3. Write an equation that has a slope of ½ and passes through the origin. The origin is (0,0). Therefore the y-intercept for this equation is 0. Y = mx+b m = ½ b = 0 Y = 1/2x + 0 or Y = 1/2x 4. A day care center charges $9 per hour plus a $3 snack fee for a Parents Night Out Special. Write an equation that can be used to determine the price of sending a child to the day care center for x number of hours. Slope is the rate and the key word is usually per So, 9 is the slope ($9 per hour) The y-intercept is a constant. $3 is the fee for snack. This is a constant in the problem. Y = mx+b m = 9 b = 3 Y = 9x + 3 is the equation that represents this problem. Suppose a parent decides to send their child to the day care center for 5 hours. How much will this cost? Justify your answer. Since x represents the number of hours, we will substitute 5 for x in the equation. Y = 9x+3 Y = 9(5)+3 Y = 48 Sending a child to the day care for 5 hours costs $48. Part 2: Writing Equations in Standard Form (Lesson 2) 5. Rewrite the following equations in standard form. a. y = -4x 9 b. y = 6x + 2 Standard Form: Ax + By = C Standard Form: Ax + By = C 4x + y = -4x+4x 9 Add 4x to both sides -6x+y = 6x-6x + 2 Subtract 6x from both sides 4x+y = -9 Simplify -6x + y = 2 Simplify The lead coefficient is positive and there are no fractions or decimals, so this is standard form. -1(-6x+y) =2(-1) 6x y = -2 Multiply by -1 to make the lead coefficient positive. Standard form

8 6. Which term in the equation does not have an integer coefficient? 1/2x + 3y = 9 An integer is a positive or negative whole number: ( -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ) Unit 3: Writing Equations The set of integers does not contain fractions or decimals. Therefore, ½ is not an integer. The term that does not have an integer coefficient is 1/2x. In order for this equation to be written in standard form, we must rewrite with integer coefficients. We must get rid of the fraction, ½. In order to do this we will multiply ALL terms by 2. 2(1/2x+3y) = 9(2) x+ 6y = 18 There are no fractions or decimals, and the lead coefficient is positive. This is now written in standard form. 8. Write the following equation in standard form: 1/3x + 1/2y = 6 This equation has 2 fractions, with different denominators. Therefore, in order to get rid of both fractions, I must multiply ALL terms by the lowest common multiple (LCM). The LCM is 6. 6(1/3x+ 1/2y) = 6(6) 2x+ 3y = 36 is the proper standard form equation.

9 9. Write an equation, in standard form, for the line shown on the graph. a. b. Unit 3: Writing Equations Since the slope and y-intercept are identifiable, it would be easiest to first write the equation in slope intercept form, then rewrite it in standard form. Y = mx+ b m = -2 b = 5 Y = -2x+5 Now rewrite in standard form: 2x+y = -2x+2x +5 2x+y = 5 Equation in slope intercept form Add 2x Simplify 2x+y = 5 is the equation in standard form. Since the slope and y-intercept are identifiable, it would be easiest to first write the equation in slope intercept form, then rewrite it in standard form. Y = mx+ b m = 4/5 b = -9 Y = 4/5x - 9 Now rewrite in standard form: Equation in slope intercept form 5y = 5(4/5x-9) Multiply by 5 5y = 4x 45 Simplify -4x + 5y 4x-4x-45 Subtract 4x -4x +5y =-45 Simplify -1(-4x+5y) = -45(-1) Multiply by -1 to make lead coefficient positive. 4x-5y = 45 is the equation in standard form.

10 Part 3: Standard Form Word Problems (Lesson 3) 10. A new 3-D movie is out and the theatre is charging $10.50 for adults and $7 for children. You have $50 to spend on the movies. Write an equation in standard form that represents the number of adult, a and child, c tickets that you can purchase. Price for adults(# of adults) + Price per child(# of children) = Total 10.50a + 7c = 50 is the equation that represents the number of tickets that you can purchase. Suppose you are taking 3 adults, at most, how many children can you take to the movies? Justify your answer mathematically. If you are taking 3 adults, then you will substitute 3 for a in your equation from above. Then solve for c a + 7c= 50 Original Equation 10.50(3) + 7c = 50 Substitute 3 for a since there are 3 adults c = 50 Simplify: 10.50(3) = c = Subtract from both sides 7c = Simplify: = c/7 = 18.50/7 Divide by 7 on both sides C= 2.64 Since you can t take part of a child to the movies, you will be able to take 2 children to the movies if you take 3 adults. Part 4: Writing Equations Given Slope and a Point (Lesson 4) 11. Write an equation for the line that has a slope of -3 and passes through the point (5,-7). We know: m = -3 x = 5 y = -7 we need to find b in order to write an equation in slope intercept form. Let s substitute what we know and solve for b. Y = mx=b -7 = -3(5) + b Substitute for m, x, and y -7 = b Simplify: -3(5) = = b Add 15 to both sides 8 = b The y-intercept (b) = 8 Now that we know: m = -3 and b = 8, we can write an equation in slope intercept form: Y = mx+ b y = -3x+ 8

11 12. A local tax service charges $65.25 an hour plus a filing fee. A three hour session costs $ Write an equation that can be used to find the total cost for any session. In this problem, we know the slope is: because this is the rate per hour. We also know that a 3 hour session is This is an ordered pair, because 3 hours is directly related to (3, ). So, we know: m = x = 3 y = Now we need to solve for b. Y=mx+b = 65.25(3) + b Substitute for m, x, and y = b Simplify: 65.25(3) = = b Subtract from both sides = b The y-intercept is Now, write an equation using the slope and y-intercept. Y =mx+b Y = 65.25x This is the equation that could be used to find the total cost for any session. Part 5: Writing Equations Given Two Points (Lesson 5) 13. Write an equation for the line that passes through the points: (6,-2) (-4, 8) Since we are given two points, we must find the slope and y-intercept. We will first find the slope using the slope formula. Then we will use the slope and 1 point to find the y-intercept. y 2 - y 1 = 8-(-2) = 10 The slope is -1 x 2 x Now we will use the slope and 1 point to find the y-intercept. Let s use (6, -2) Y= mx+ b m = -1 x = 6 y = -2 2 = -1(6) + b Substitute for m, x, and y 2 = -6 + b Simplify: (-1)(6) = = b Add 6 to both sides 8 = b The y-intercept = 8 Y = mx+ b m = -1 b = 8 Y = -x + 8 is the equation that passes through the points (6,-2) & (-4,8)

12 14. In 1991,the cost of tuition for a private school was $5,000 per year. In 2012, the cost of the same private school is $12,500. Let x = 0 represent the year Write an equation that could be used to predict the cost of tuition for any given year. In this problem, we know that in the year 1991 the cost was $5000. Since these two are directly related, this is an ordered pair. (1991, 5000). But, since x = 0 represents 1990, x = 1 represents 1991 since its one year later, so our actual ordered pair is (1, 5000) In the year 2012, the cost was $12,500. Again this is a direct relationship, so it s an ordered pair. (2012, 12500). Since = 0 represents 1990, x = 22 represents 2012 since its 22 years after (22, 12500) Now that we have two ordered pairs: (1, 5000) & (22, 12500) we can find the slope using the slope formula. y 2 - y 1 = = 7500 = x 2 x The slope is and 1 point is (1, 5000). Let s us this to find the y-intercept (b). Y = mx+ b 5000 = (1) + b Substitute for m, x, and y = b Simplify: (1) = = b Subtract from both sides = b The y-intercept = Now use the slope and y intercept to write the equation. m = b = Y = mx+ b Y = Predict what the tuition will be for the year Now using our new equation, we can predict what the tuition will be for the year Y = x Y = (25) Substitute 25 for x since 2015 is 25 years beyond Y = In the year 2015, the tuition for this school will be about $

13 15. In 2003, the cost of season football tickets was $2,200. In 2012, the cost of the same season tickets is $4,550. Let x = 0 represent the year Write an equation that could be used to predict the cost of the tickets for any given year. In 2003, the cost of tickets was $2200. This is an ordered pair. Since x = 0 represents 2000, our ordered pair is (3, 2200) is three years later than In 2012 the tickets were $4550. This is also an ordered pair. (12, 4550) 2012 is 12 years later than Now we will use our ordered pairs to find the slope for this equation. (3, 2200) (12, 4550) y 2 - y 1 = = 2350 = The slope is x 2 x Now we will use the slope and 1 point (3, 2200) to find the y-intercept. Y = mx+b m = x = 3 y = = 261.1(3) + b Substitute for m, x, and y = b Simplify: 261.1*3 = = b Subtract from both sides = b The y-intercept is Now that we know the slope and y-intercept, we can write our equation. Y = mx+ b Y = 261.1x is the equation that can be used to predict the cost of the tickets. Predict what the cost of the tickets will be in the year Y = 261.1x Equation from above Y = 261.1(20) Substitute 20 for x since 2020 is 20 years after Y = Approximate cost of tickets. The tickets will cost approximately $ in the year 2020.

14 Part 6: Writing Linear Equations - Point-Slope Form (Lesson 6) 16. Write an equation that has a slope of -3 and passes through the point (2,8) Since we are given slope and a point, we can use point-slope to write the equation. m = -3, x 1 = 2 y 1 =8 y y 1 = m(x-x 1 ) y 8 = -3(x-2) Substitute the values into the equation y 8 = -3x +6 Distribute -3 y 8 +8 = -3x Add 8 to both sides y = -3x +14 The equation that has a slope of -3 and passes through (2,8) 17. Write an equation that passes through the points (1,9) and (3, -8). Explain the steps that you used to solve this problem. Since we are given two points, we must first start by finding the slope of the line that passes through the two points. We will use the slope formula to find the slope. = = 17 2 Now we can use the point-slope form to write the equation. (*You can use either point for x 1 and y 1 ) m = -17/2, x 1 = 1 y 1 =9 y y 1 = m(x-x 1 ) y 9 = -17/2(x-1) Substitute the values into the equation. y-9 = -17/2x +17/2 Distribute the -17/2 throughout the parenthesis y-9+9 = -17/2x +17/2 +9 Add 9 to both sides y = -17/2x+ 35/2 18. Write an equation that has a slope of 1/2 and an x-intercept of 1. Explain how you solved this problem. We know m = ½ and an x-intercept of 1 means that we have the point (1,0). Use point slope form. m = 1/2, x 1 = 1 y 1 =0 y y 1 = m(x-x 1 ) y 0 = 1/2(x-1) y = 1/2x 1/2 Substitute the values into the equation. 19. Write an equation that has a y-intercept of 22 and passes through the point (3, -7). We know the y-intercept is 22, but we don t know the slope which is essential. If we write the y intercept as a point: (0,22) and we use the other point (3, -7) you can use the slope formula to find the slope. = = M = -29/3 b = 22 is the slope. y = -29/3x +22 is the equation.

15 20. Write an equation that has an x-intercept of -5 and a y-intercept of 8. Since we are given the x and y intercept, we can write and equation in standard form pretty easily. We know that a common multiple of 5 and 8 is 40, so that will be our constant. Since we know (-5,0) & (0,8) are intercepts, we need to write an equation that works: A(-5) + B(8) = 40 In order for the x and y intercepts to work, A = -8 and B = 5-8x + 5y = 40. Or 8x 5y = -40 if you want to make the lead coefficient work. Justify: 8x 5y = -40 substitute (-5,0) and (0,8) and make sure they work. 8(-5) 5(0) = -40 8(0) 5(8) = = =-40 It works! *You can also write an equation in slope intercept form. If you took this route, your equation would be: Y = 8/5x Mei earned $38 on her investment in 5 months. She earned $62 on the same investment in 8 months. Write an equation that can be used to find the amount earned (y) on Mei s invest in x number of months. We are given two points in this problem: (5, 38) (8, 62). We can find the slope and then use point-slope form to write the equation. = = 24 3 =8 Now we can use the point-slope form to write the equation. (*You can use either point for x 1 and y 1 ) m = 8, x 1 = 5 y 1 =38 y y 1 = m(x-x 1 ) y 38 = 8(x-5) Substitute the slope and point coordinates into the equation. y 38 = 8x 40 Distribute 8 throughout the parenthesis. Y = 8x Add 38 to both sides of the equation. Y = 8x -2 About how much will Mei earn after 15 months? After 15 months, Mei will have earned $118. Y = 8x 2 Y = 8(15) 2 Y = 118

16 About how much is Mei earning per month? Explain how you determined your answer. Mei is earning about $8 per month on her investment. Part 7: Parallel and Perpendicular Lines (Lesson 7) Parallel Lines: Have the same slope and different y-intercepts. Perpendicular lines: The slopes are the negative reciprocal of each other. 22. Write an equation for a line that passes through (2,-8) and is perpendicular to a line whose slope is 5. Explain how you determined your answer. Since a line that is perpendicular will have a slope that is the negative reciprocal, that means that the slope of this line will be -1/5. Now we can use point slope form to write the equation of the line. m = -1/5, x 1 = 2 y 1 =-8 y y 1 = m(x-x 1 ) y -8 = -1/5(x-2) Substitute values into the equation. y+8 = -1/5x +2/5 Distribute -1/5 y = -1/5x +2/5 8 Subtract 8 from both sides of the equation. y = -1/5x 38/5 is the line perpendicular. 23. Write an equation for a line that passes through (-3,6) and is parallel to the graph of x = 5. X = 5 is a vertical line. Therefore, a line that is parallel is also going to be a vertical line. If it passes through (-3,6), then it must be a vertical line through (-3,6) which means that the equation is: x = -3

17 24. Write an equation for a line that is perpendicular to the graph of 2x 2y = 10 and intersects the graph at its x-intercept. We must first determine the slope of 2x-2y = 10 by rewriting it in slope intercept form. 2x 2x 2y = -2x +10 Subtract 2x from both sides -2y = -2x +10-2y/-2 = -2x/-2 +10/-2 Divide all terms by -2 Y = x -5 The slope of this line is 1 Now we can use point-slope form. Slope will be -1 for the perpendicular line and passes through the x- intercept which is 5 or (5,0). 2x = 10 X = 5 m = -1 x 1 = 5 y 1 =-0 y y 1 = m(x-x 1 ) y 0 = -1(x-5) Substitute the given values. Y = -x +5 Is the equation for a line perpendicular to 2x-27 = 10 and passes through its x-intercept. Part 8: Scatter Plots and Line of Best Fit (Lesson 8) 25. The following data represents the total U.S. E-commerce sales from (Statistic from statista.com) Let x =0 represent the year Year E-commerce Sales (in billions of dollars) Using your graphing calculator, write an equation for the line of best fit for this data. Line of best fit is: y = 20.45x (all numbers in billions of dollars) What does the slope and y-intercept represent in the context of this problem.? In this problem, the slope represents the amount that the E-commerce sales increase per year. The E-commerce sales increase by billion dollars each year. The y-intercept represents the amount of E-commerce sales in the year 2000 because when x = 0 this represents the year Therefore, the estimated amount of E-commerce sales in 2000 was $37.7 billion dollars.

18 Predict the amount of E-commerce sales in the U.S. for the year Explain how you determined your answer. In order to predict the amount for the year 2012, we must substitute 12 into our line of best fit equation. y = 20.45x y = 20.45(12) y = billion E-commerce sales in 2012 will be about $283.1 billion dollars.