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Transcript
January 26, 2017 Learning Target: Continue work with solving systems of equations using the Equal Values Method when equations are not in y-form and learn to identify systems that represent the same line or parallel lines (that is, systems that have infinitely many solutions or no solution). Focus Questions • What is a solution to an equation? What does it look like? • What is the growth pattern? • What is the y-intercept? January 26, 2017 C Think, Ink, Pair, Share You have been introduced to systems of linear equations that are used to represent various situations. You have used the Equal Values Method to solve systems algebraically. Just as in linear equations you found that sometimes there were no solutions or an infinite number of solutions, today you will discover how this same situation occurs for systems of equations. 5-52. Sara has agreed to help with her younger sister’s science fair experiment. Her sister planted string beans in two pots. She is using a different fertilizer in each pot to see which one will grow the tallest plant. Currently, plant A is 4 inch per day, while plant B is 9 inches tall and grows inches tall and grows inch per day. If the plants continue growing at these rates, in how many days will the two plants be the same height? Which plant will be tallest in six weeks? Write a system of equations and solve. January 26, 2017 Think, Ink, Pair, Share C 5-53. Felipe applied for a job. The application process required him to take a test of his math skills. One problem on the test was a system of equations, but one of the equations not in y = mx + b form. The two equations are shown below. 3x + 2y = 9 y= x-5 Work with your team to find a way to solve the equations using the Equal Values Method. January 26, 2017 Vocab Tool Kit pg. 43 coefficient (numerical) A number multiplying a variable or product of variables. For example, −7 is the coefficient of −7x. constant A number that is not multiplied by a variable. In the expression 2x + 3(5 − 2x) + 8, the number 8 is a constant term. The number 3 is not a constant term, because it is multiplied by a variable inside the parentheses. solution The number or numbers that when substituted into an equation or inequality make the equation or inequality true. For example, x = 4 is a solution to the equation 3x − 2 = 10 because 3x − 2 equals 10 when x = 4 January 26, 2017 Solutions to a System of Equations A solution to a system of equations gives a value for each variable that makes both equations true. For example, when 4 is substituted for x and 5 is substituted for y in both equations at right, both equations are true. So x = 4 and y = 5 or (4, 5) is a solution to this system of equations. When the two equations are graphed, (4, 5) is the point of intersection. System with one solution: Some systems of equations have one solution. When the lines are graphed they intersect at one point. Some systems have no solution. Notice that the Equal Values Method would yield 3 = 4 , which is never true. When the lines are graphed, they are parallel. Some systems have infinite solutions. Notice that the Equal Values Method would yield 0 = 0 , which is always true. When the lines are graphed, they overlap. intersecting lines x − y = −1 2x − y = 3 System with no solutions: parallel lines x + y = 3 x + y = 4 System with infinite solutions: coinciding lines x + y = 3 2x + 2y = 6