Exercises
\nSolve each of the following equations for x.<\/p>\n
Exercise 1.
\n7x-3=5x+5
\nAnswer:
\n7x-3=5x+5
\n7x-3+3=5x+5+3
\n7x=5x+8
\n7x-5x=5x-5x+8
\n2x=8
\nx=4<\/p>\n
Exercise 2.
\n7x-3=7x+5
\nAnswer:
\n7x-3=7x+5
\n7x-7x-3=7x-7x+5
\n-3\u22605
\nThis equation has no solution.<\/p>\n
Exercise 3.
\n7x-3=-3+7x
\nAnswer:
\n7x-3=-3+7x
\n7x-3+3=-3+3+7x
\n7x=7x
\nOR
\n7x-3=-3+7x
\n7x-7x-3=-3+7x-7x
\n-3=-3<\/p>\n
Exercises 4\u201310<\/p>\n
Give a brief explanation as to what kind of solution(s) you expect the following linear equations to have. Transform the equations into a simpler form if necessary.<\/p>\n
Exercise 4.
\n11x-2x+15=8+7+9x
\nAnswer:
\nIf I use the distributive property on the left side, I notice that the coefficients of the x are the same, specifically 9, and when I simplify the constants on the right side, I notice that they are the same. Therefore, this equation has infinitely many solutions.<\/p>\n
Exercise 5.
\n3(x-14)+1=-4x+5
\nAnswer:
\nIf I use the distributive property on the left side, I notice that the coefficients of x are different. Therefore, the equation has one solution.<\/p>\n
Exercise 6.
\n-3x+32-7x=-2(5x+10)
\nAnswer:
\nIf I use the distributive property on the each side of the equation, I notice that the coefficients of x are the same, but the constants are different. Therefore, this equation has no solutions.<\/p>\n
Exercise 7.
\n\\(\\frac{1}{2}\\) (8x+26)=13+4x
\nAnswer:
\nIf I use the distributive property on the left side, I notice that the coefficients of x are the same, specifically 4, and the constants are also the same, 13. Therefore, this equation has infinitely many solutions.<\/p>\n
Exercise 8.
\nWrite two equations that have no solutions.
\nAnswer:
\nAnswers will vary. Verify that students have written equations where the coefficients of x on each side of the equal sign are the same and that the constants on each side are unique.<\/p>\n
Exercise 9.
\nWrite two equations that have one unique solution each.
\nAnswer:
\nAnswers will vary. Accept equations where the coefficients of x on each side of the equal sign are unique.<\/p>\n
Exercise 10.
\nWrite two equations that have infinitely many solutions.
\nAnswer:
\nAnswers will vary. Accept equations where the coefficients of x and the constants on each side of the equal sign are the same.<\/p>\n
Students apply their knowledge of solutions to linear equations by writing equations with unique solutions, no solutions, and infinitely many solutions.<\/p>\n
Question 1.
\nGive a brief explanation as to what kind of solution(s) you expect for the linear equation 18x+\\(\\frac{1}{2}\\) =6(3x+25). Transform the equation into a simpler form if necessary.
\nAnswer:
\nIf I use the distributive property on the right side of the equation, I notice that the coefficients of x are the same, but the constants are different. Therefore, this equation has no solutions.<\/p>\n
Question 2.
\nGive a brief explanation as to what kind of solution(s) you expect for the linear equation 8-9x=15x+7+3x. Transform the equation into a simpler form if necessary.
\nAnswer:
\nIf I collect the like terms on the right side of the equation, then I notice that the coefficients of x are different, and so are the constants. Therefore, this equation will have a unique solution.<\/p>\n
Question 3.
\nGive a brief explanation as to what kind of solution(s) you expect for the linear equation 5(x+9)=5x+45. Transform the equation into a simpler form if necessary.
\nAnswer:
\nThis is an identity under the distributive property. Therefore, this equation will have infinitely many solutions.<\/p>\n
Question 4.
\nGive three examples of equations where the solution will be unique; that is, only one solution is possible.
\nAnswer:
\nAccept equations where the coefficients of x on each side of the equal sign are unique.<\/p>\n
Question 5.
\nSolve one of the equations you wrote in Problem 4, and explain why it is the only solution.
\nAnswer:
\nVerify that students solved one of the equations. They should have an explanation that includes the statement that there is only one possible number that could make the equation true. They may have referenced one of the simpler forms of their transformed equation to make their case.<\/p>\n
Question 6.
\nGive three examples of equations where there will be no solution.
\nAnswer:
\nAccept equations where the coefficients of x on each side of the equal sign are the same, and the constants on each side are unique.<\/p>\n
Question 7.
\nAttempt to solve one of the equations you wrote in Problem 6, and explain why it has no solution.
\nAnswer:
\nVerify that students have solved one of the equations. They should have an explanation that includes the statement about getting a false equation (e.g., 6\u226010).<\/p>\n
Question 8.
\nGive three examples of equations where there will be infinitely many solutions.
\nAnswer:
\nAccept equations where the coefficients of x and constants on each side of the equal sign are the same.<\/p>\n
Question 9.
\nAttempt to solve one of the equations you wrote in Problem 8, and explain why it has infinitely many solutions.
\nAnswer:
\nVerify that students have solved one of the equations. They should have an explanation that includes the statement about the linear expressions being exactly the same, an identity; therefore, any rational number x would make the equation true.<\/p>\n
Give a brief explanation as to what kind of solution(s) you expect the following linear equations to have. Transform the equations into a simpler form if necessary.<\/p>\n
Question 1.
\n3(6x+8)=24+18x
\nAnswer:
\nIf I use the distributive property on the left side, I notice that the coefficients of x are the same, and the constants are the same. Therefore, this equation has infinitely many solutions.<\/p>\n
Question 2.
\n12(x+8)=11x-5
\nAnswer:
\nIf I use the distributive property on the left side, I notice that the coefficients of x are different, and the constants are different. Therefore, this equation has a unique solution.<\/p>\n
Question 3.
\n5x-8=11-7x+12x
\nAnswer:
\nIf I collect the like terms on the right side, I notice that the coefficients of x are the same, but the constants are different. Therefore, this equation has no solution.<\/p>\n","protected":false},"excerpt":{"rendered":"
Engage NY Eureka Math 8th Grade Module 4 Lesson 7 Answer Key Eureka Math Grade 8 Module 4 Lesson 7 Exercise Answer Key Exercises Solve each of the following equations for x. Exercise 1. 7x-3=5x+5 Answer: 7x-3=5x+5 7x-3+3=5x+5+3 7x=5x+8 7x-5x=5x-5x+8 2x=8 x=4 Exercise 2. 7x-3=7x+5 Answer: 7x-3=7x+5 7x-7x-3=7x-7x+5 -3\u22605 This equation has no solution. Exercise … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/85056"}],"collection":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/comments?post=85056"}],"version-history":[{"count":1,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/85056\/revisions"}],"predecessor-version":[{"id":137230,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/posts\/85056\/revisions\/137230"}],"wp:attachment":[{"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/media?parent=85056"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/categories?post=85056"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ccssanswers.com\/wp-json\/wp\/v2\/tags?post=85056"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}