{"id":86300,"date":"2021-04-07T14:12:55","date_gmt":"2021-04-07T08:42:55","guid":{"rendered":"https:\/\/ccssmathanswers.com\/?p=86300"},"modified":"2022-04-01T10:25:45","modified_gmt":"2022-04-01T04:55:45","slug":"eureka-math-grade-8-module-4-lesson-18","status":"publish","type":"post","link":"https:\/\/bigideasmathanswers.com\/eureka-math-grade-8-module-4-lesson-18\/","title":{"rendered":"Eureka Math Grade 8 Module 4 Lesson 18 Answer Key"},"content":{"rendered":"

Engage NY Eureka Math 8th Grade Module 4 Lesson 18 Answer Key<\/h2>\n

Eureka Math Grade 8 Module 4 Lesson 19 Opening Exercise Answer Key<\/h3>\n

Examine each of the graphs and their equations. Identify the coordinates of the point where the line intersects the y-axis. Describe the relationship between the point and the equation y=mx+b.
\na. y=\\(\\frac{1}{2}\\) x+3
\n\"Eureka
\nb. y=-3x+7
\n\"Eureka
\nc. y=-\\(\\frac{2}{3}\\) x-2
\n\"Eureka
\nd. y=5x-4
\n\"Eureka
\nAnswer:
\nA point is noted in each graph above where the line intersects the y-axis:
\ny=\\(\\frac{1}{2}\\) x+3, (0,3)
\ny=-3x+7, (0,7)
\ny=-\\(\\frac{2}{3}\\) x-2, (0,-2)
\ny=5x-4, (0,-4)
\nIn each equation, the number b was the y-coordinate of the point where the line intersected the y-axis.<\/p>\n

Eureka Math Grade 8 Module 4 Lesson 19 Example Answer Key<\/h3>\n

Graph an equation in the form of y=mx+b.<\/p>\n

Example 1.
\nGraph the equation y=\\(\\frac{2}{3}\\) x+1. Name the slope and y-intercept point.
\nAnswer:
\nThe slope is m=\\(\\frac{2}{3}\\), and the y-intercept point is (0,1).<\/p>\n

\u2192 To graph the equation, we must begin with the known point. In this case, that is the y-intercept point. We cannot begin with the slope because the slope describes the rate of change between two points. That means we need a point to begin with. On a graph, we plot the point (0,1).
\n\"Engage<\/p>\n

\u2192 Next, we use the slope to find the second point. We know that m=\\(\\frac{|Q R|}{|P Q|}\\) =\\(\\frac{2}{3}\\). The slope tells us exactly how many units to go to the right of P to find point Q and then how many vertical units we need to go from Q to find point R. How many units will we go to the right in order to find point Q? How do you know?
\n\u2192 We need to go 3 units to the right of point P to find Q. We go 3 units because |PQ|=3.
\n\"Engage
\n\u2192 How many vertical units from point Q must we go to find point R? How do you know?
\n\u2192 We need to go 2 units from point Q to find R. We go 2 units because |QR|=2.
\n\u2192 Will we go up from point Q or down from point Q to find R? How do you know?
\n\u2192 We need to go up because the slope is positive. That means that the line will be left-to-right inclining.
\n\"Engage
\n\u2192 Since we know that the line joining two distinct points of the form y=mx+b has slope m, and we specifically constructed points P and R with the slope in mind, we can join the points with a line.
\n\"Engage<\/p>\n

Example 2.
\nGraph the equation y=-\\(\\frac{3}{4}\\) x-2. Name the slope and y-intercept point.
\nAnswer:
\nThe slope is m=-\\(\\frac{3}{4}\\), and the y-intercept point is (0,-2).<\/p>\n

\u2192 How do we begin?
\n\u2192 We must begin by putting a known point on the graph, (0,-2).
\n\"Engage<\/p>\n

\u2192 We know that m=\\(\\frac{|Q R|}{|P Q|}\\) =-\\(\\frac{3}{4}\\). How many units will we go to the right in order to find point Q? How do you know?
\n\u2192 We need to go 4 units to the right of point P to find Q. We go 4 units because |PQ|=4.
\n\"Engage
\n\u2192 How many units from point Q must we go to find point R? How do you know?
\n\u2192 We need to go 3 units from point Q to find R. We go 3 units because |QR|=3.
\n\u2192 Will we go up from point Q or down from point Q to find R? How do you know?
\n\u2192 We need to go down from point Q to point R because the slope is negative. That means that the line will be left-to-right declining.
\n\"Engage
\n\u2192 Now we draw the line through the points P and R.
\n\"Engage<\/p>\n

Example 3
\nGraph the equation y=4x-7. Name the slope and y-intercept point.
\nAnswer:
\nThe slope is m=4, and the y-intercept point is (0,-7).<\/p>\n

\u2192 Graph the equation y=4x-7. Name the slope and y-intercept point.
\n\u2192 The slope is m=4, and the y-intercept point is (0,-7).
\n\u2192 How do we begin?
\n\u2192 We must begin by putting a known point on the graph, (0,-7).
\n\"Engage
\n\u2192 Notice this time that the slope is the integer 4. In the last two examples, our slopes have been in the form of a fraction so that we can use the information in the numerator and denominator to determine the lengths of |PQ| and |QR|. Since m=\\(\\frac{|Q R|}{|P Q|}\\) =4, what fraction can we use to represent slope to help us graph?
\n\u2192 The number 4 is equivalent to the fraction \\(\\frac{4}{1}\\).
\n\u2192 Using m=\\(\\frac{|Q R|}{|P Q|}\\) =\\(\\frac{4}{1}\\), how many units will we go to the right in order to find point Q? How do you know?
\n\u2192 We need to go 1 unit to the right of point P to find Q. We go 1 unit because |PQ|=1.
\n\"Engage
\n\u2192 How many vertical units from point Q must we go to find point R? How do you know?
\n\u2192 We need to go 4 units from point Q to find R. We go 4 units because |QR|=4.
\n\u2192 Will we go up from point Q or down from point Q to find R? How do you know?
\n\u2192 We need to go up from point Q to point R because the slope is positive. That means that the line will be left-to-right inclining.
\n\"Engage<\/p>\n

\u2192 Now we join the points P and R to make the line.
\n\"Engage<\/p>\n

Eureka Math Grade 8 Module 4 Lesson 19 Exercise Answer Key<\/h3>\n

Exercises 1\u20134 (5 minutes)
\nStudents complete Exercises 1\u20134 individually or in pairs.<\/p>\n

Exercise 1.
\nGraph the equation y=\\(\\frac{5}{2}\\) x -4.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=\\(\\frac{5}{2}\\), and the y-intercept point is (0,-4).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Exercise 2.
\nGraph the equation y=-3x+6.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=-3, and the y-intercept point is (0,6).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 3.
\nThe equation y=1x+0 can be simplified to y=x. Graph the equation y=x.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=1, and the y-intercept point is (0,0).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 4.
\nGraph the point (0,2).
\nAnswer:
\n\"Eureka<\/p>\n

a. Find another point on the graph using the slope, m=\\(\\frac{2}{7}\\).<\/p>\n

b. Connect the points to make the line.<\/p>\n

c. Draw a different line that goes through the point (0,2) with slope m=\\(\\frac{2}{7}\\). What do you notice?
\nAnswer:
\nOnly one line can be drawn through the given point with the given slope.<\/p>\n

Exercises 5\u20136
\nStudents complete Exercises 5\u20136 individually or in pairs.<\/p>\n

Exercise 5.
\nA bank put $10 into a savings account when you opened the account. Eight weeks later, you have a total of $24. Assume you saved the same amount every week.
\na. If y is the total amount of money in the savings account and x represents the number of weeks, write an equation in the form y=mx+b that describes the situation.
\nAnswer:
\n24=m(8)+10
\n14=8m
\n\\(\\frac{14}{8}\\)=m
\n\\(\\frac{7}{4}\\)=m
\ny=\\(\\frac{7}{4}\\) x+10<\/p>\n

b. Identify the slope and the y-intercept point. What do these numbers represent?
\nAnswer:
\nThe slope is \\(\\frac{7}{4}\\), and the y-intercept point is (0,10). The y-intercept point represents the amount of money the bank gave me, in the amount of $10. The slope represents the amount of money I save each week, \\(\\frac{7}{4}\\)=$1.75.<\/p>\n

c. Graph the equation on a coordinate plane.
\nAnswer:
\n\"Eureka<\/p>\n

d. Could any other line represent this situation? For example, could a line through point (0,10) with slope \\(\\frac{7}{5}\\) represent the amount of money you save each week? Explain.
\nAnswer:
\nNo, a line through point (0,10) with slope \\(\\frac{7}{5}\\) cannot represent this situation. That line would show that at the end of the 8 weeks I would have $21.20, but I was told that I would have $24 by the end of the 8 weeks.<\/p>\n

Exercise 6.
\nA group of friends are on a road trip. After 120 miles, they stop to eat lunch. They continue their trip and drive at a constant rate of 50 miles per hour.
\na. Let y represent the total distance traveled, and let x represent the number of hours driven after lunch. Write an equation to represent the total number of miles driven that day.
\nAnswer:
\ny=50x+120<\/p>\n

b. Identify the slope and the y-intercept point. What do these numbers represent?
\nAnswer:
\nThe slope is 50 and represents the rate of driving. The y-intercept point is 120 and represents the number of miles they had already driven before driving at the given constant rate.<\/p>\n

c. Graph the equation on a coordinate plane.
\nAnswer:
\n\"Eureka<\/p>\n

d. Could any other line represent this situation? For example, could a line through point (0,120) with slope 75 represent the total distance the friends drive? Explain.
\nAnswer:
\nNo, a line through point (0,120) with a slope of 75 could not represent this situation. That line would show that after an hour, the friends traveled a total distance of 195 miles. According to the information given, the friends would only have traveled 170 miles after one hour.<\/p>\n

Eureka Math Grade 8 Module 4 Lesson 19 Exit Ticket Answer Key<\/h3>\n

Mrs. Hodson said that the graphs of the equations below are incorrect. Find the student\u2019s errors, and correctly graph the equations.<\/p>\n

Question 1.
\nStudent graph of the equation y=\\(\\frac{1}{2}\\) x+4:
\n\"Engage<\/p>\n

Error:
\nAnswer:
\nThe student should have gone up 1 unit when finding |QR| since the slope is positive.
\nCorrect graph of the equation:
\n\"Engage
\nAnswer:
\n\"Engage<\/p>\n

Question 2.
\nStudent graph of the equation y=-\\(\\frac{3}{5}\\) x-1:
\n\"Engage<\/p>\n

Error:
\nAnswer:
\nThe student did not find the y-intercept point correctly. It should be the point (0,-1).
\nCorrect graph of the equation:
\n\"Engage
\nAnswer:
\n\"Engage<\/p>\n

Eureka Math Grade 8 Module 4 Lesson 19 Problem Set Answer Key<\/h3>\n

Students practice graphing equations using y-intercept point and slope. Students need graph paper to complete the Problem Set. Optional Problem 11 has students show that there is only one line passing through a point with a given negative slope.<\/p>\n

Graph each equation on a separate pair of x- and y-axes.<\/p>\n

Question 1.
\nGraph the equation y=\\(\\frac{4}{5}\\) x-5.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=\\(\\frac{4}{5}\\), and the y-intercept point is (0,-5).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 2.
\nGraph the equation y=x+3.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=1, and the y-intercept point is (0,3).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 3.
\nGraph the equation y=-\\(\\frac{4}{3}\\) x+4.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=-\\(\\frac{4}{3}\\), and the y-intercept point is (0,4).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 4.
\nGraph the equation y=\\(\\frac{5}{2}\\) x.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=\\(\\frac{5}{2}\\), and the y-intercept point is (0,0).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 5.
\nGraph the equation y=2x-6.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=2, and the y-intercept point is (0,-6).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 6.
\nGraph the equation y=-5x+9.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=-5, and the y-intercept point is (0,9).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 7.
\nGraph the equation y=\\(\\frac{1}{3}\\) x+1.
\na. Name the slope and the y-intercept point.
\nAnswer:
\nThe slope is m=\\(\\frac{1}{3}\\), and the y-intercept point is (0,1).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 8.
\nGraph the equation 5x+4y=8. (Hint: Transform the equation so that it is of the form y=mx+b.)
\na. Name the slope and the y-intercept point.
\nAnswer:
\n5x+4y=8
\n5x-5x+4y=8-5x
\n4y=8-5x
\n\\(\\frac{4}{4}\\) y=\\(\\frac{8}{4}\\)–\\(\\frac{5}{4}\\) x
\ny=2-\\(\\frac{5}{4}\\) x
\ny=-\\(\\frac{5}{4}\\) x+2
\nThe slope is m=-\\(\\frac{5}{4}\\), and the y-intercept point is (0,2).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 9.
\nGraph the equation -2x+5y=30.
\na. Name the slope and the y-intercept point.
\nAnswer:
\n-2x+5y=30
\n2x+2x+5y=30+2x
\n5y=30+2x
\n\\(\\frac{5}{5}\\) y=\\(\\frac{30}{5}\\)+\\(\\frac{2}{5}\\) x
\ny=6+\\(\\frac{2}{5}\\) x
\ny=\\(\\frac{2}{5}\\) x+6
\nThe slope is m=\\(\\frac{2}{5}\\), and the y-intercept point is (0,6).<\/p>\n

b. Graph the known point, and then use the slope to find a second point before drawing the line.
\nAnswer:
\n\"Eureka<\/p>\n

Question 10.
\nLet l and l’ be two lines with the same slope m passing through the same point P. Show that there is only one line with a slope m, where m<0, passing through the given point P. Draw a diagram if needed.
\nAnswer:
\nFirst, assume that there are two different lines l and l’ with the same negative slope passing through P. From point P, I mark a point Q one unit to the right. Then, I draw a line parallel to the y-axis through point Q. The intersection of this line and line l and l’ are noted with points R and R’, respectively. By definition of slope, the lengths |QR| and |QR’ | represent the slopes of lines l and l’, respectively. We are given that the lines have the same slope, which means that lengths |QR| and |QR’| are equal. Since that is true, then points R and R’coincide and so do lines l and l’. Therefore, our assumption that they are different lines is false; l and l’ must be the same line. Therefore, there is only one line with slope m passing through the given point P.
\n\"Eureka<\/p>\n","protected":false},"excerpt":{"rendered":"

Engage NY Eureka Math 8th Grade Module 4 Lesson 18 Answer Key Eureka Math Grade 8 Module 4 Lesson 19 Opening Exercise Answer Key Examine each of the graphs and their equations. Identify the coordinates of the point where the line intersects the y-axis. Describe the relationship between the point and the equation y=mx+b. a. … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[6],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/86300"}],"collection":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/comments?post=86300"}],"version-history":[{"count":1,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/86300\/revisions"}],"predecessor-version":[{"id":91385,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/posts\/86300\/revisions\/91385"}],"wp:attachment":[{"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/media?parent=86300"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/categories?post=86300"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bigideasmathanswers.com\/wp-json\/wp\/v2\/tags?post=86300"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}