System of Linear Equation: with more than Two Equations and more than Two Unknowns Michigan Department of Education Standards for High School: Standard 1: Solve linear equations and inequalities including systems of up to three linear equations with three unknowns. Justify your steps in the solution. Set: As the students are entering the classroom, have then move into the groups that you have placed them into, as indicated on the board. These groups will be based on their current ability on the math topics covered so far. Once they get into their groups they will need to begin thinking about the following situation: - The local fruit stand is owned by an eccentric mathematician who does not list prices for the fruit he sells. Instead of prices, he has a list of three equations that give the value of the fruit. Using the equations, see if you can find how many pears are equal to the value of a cantaloupe? Give then 5 minutes to think about way to solve the problem as a group and then pull them back together as a class. (Note: Make sure they stay in their groups.) Objectives: Cantaloupe 2 apples = pear 2 bananas + 2 pears = apple 2 pears = apple Students will be able to solve a system of linear equations with more than two equations and more than two unknowns. Information and Modeling #1: *The following information was taken from (Z a c c a r o, pages 203-207) 1. Going over the Set: a. Ask which groups has an idea of how to solve the problem b. As their ideas come up if it is a correct step complete it and if not have them rethink. c. One correct way is as follows: i. Using equations 1 and 3 we can replace the pears with apples and we get 2. Let s try another example: How many dogs equal a rat?
a. Step 1: Get rid of snakes b. Step 2: Subtract equation (snakes are gone) c. Step 3: Get rid of cats d. Step 4: add equation (cats are gone) 3. Remember to use the tools you learned in the previous lessons to get rid of the variables you don t want. Eventually you will be left with one variable that you need to answer the question. 4. Group Assignment a. In their groups they will receive the worksheet that corresponds to the group s ability. Worksheet are as follows: i. Worksheet A: Medium (Is a mix of two equation and three variables and three equations and three variables) ii. Worksheet B: Low (has two equations and three variables) iii. Worksheet C: High (uses 2 equations and three variables, three equations and three variables, and three equations and four variables) b. As a group the students can work together to figure out answers but should write down their own solutions and answer.
Check for Understanding #1: - The individually written worksheet will be turned in, to assess their progress using three variables and three unknowns. These worksheets will be taken for a completion/effort grade. - Although they completed the worksheet in groups, it still shows: how well they understand the material and as a group what they need help on. Information and Modeling #2: ** The following information can from (P e t e r s o n ) 1. The Definition a. A linear system of equations in three variables is made up of three equations that each contains three variables. The solution to such a system is an ordered triple, (x, y, z) that satisfies all three equations. 2. You can have three possibilities for solutions a. One unique solution, a point of intersection of all three planes. Such a system is said to be consistent. b. No solution. All three planes never intersect. Such a system is said to be inconsistent. c. Infinitely many solutions. The three planes intersect in infinitely many places. Such a system is said to be dependent. 3. As with systems of lines, if you eliminate the variables and get a true statement it means there are an infinite number of solutions. If you eliminate the variables and get a false statement it means there is no solution. 4. To solve a linear system of three equations with three variables we will use the addition method that we learned previously a. Example: i. Solving these type systems is a bit more complicated than a system of two equations so we will begin by numbering the equations. This is to help us keep track of our work. ii. Now we need to pick a variable to eliminate. There is no right choice, we are free to pick x or y or z. But, once we pick a variable we must stay with that choice. Let s choose to eliminate y. iii. We must now select two equations to add together and eliminate y. Again, there is no right choice. Let s pick equations 2 and 3. In order for the y s to cancel we must multiply equation 2 by 2.
iv. The result of this addition produces a new equation that we will number as 4. 1. Equation 4 can be simplified by dividing both sides by 5. This produces the following: v. We now repeat the process of steps 3 and 4 but with equations 1 and 2. We will need to multiply one of the equations by -1 so that the y s will cancel. It doesn t matter which one so let s choose equation number 1. vi. The result of this addition also produces a new equation that we will number as 5. vii. We now solve the system made up of equations 4 and 5. This is just a system of two equations with two variables--just like in 8.1. viii. We have the z-coordinate of the solution. By substituting -1/2 into either equation 4 or 5 we can find the x-coordinate. We will choose equation 5. ix. By substituting x = 1/2 and z = -1/2 into either equation 1, 2 or 3 we find the y- coordinate of the solution. We choose equation 2.
x. The solution is the following ordered triple. Check for understanding #2/ Guided Practice: * * P r o b l e m s f o u n d a t ( S e w a r d ). 1. Go through the following examples with the students so that they get the feeling for the homework. a. Example 1 i. You will find that if you plug the ordered triple (3/4, -2, 1/2) into ALL THREE equations of the original system, this is a solution to ALL THREE of them. ii. (3/4, -2, 1/2) is a solution to our system. b. Example 2 i. There is no value to plug in here. ii. When they end up being the same equation, you have an infinite number of solutions. You can write up your answer by writing out any of the three equations to indicate that they are the same equation. iii. Three ways to write the answer are {(x, y, z) x + y = 9} OR {(x, y, z) y + z = 7} OR {(x, y, z) x - z = 2}. c. Example 3
i. There is no value to plug in here. ii. The answer is no solution. Formative Assessment: * * P r o b l e m s f o u n d a t ( S e w a r d ). Have the student solve the following three problems. 1. 2. 3. - Each problem they must complete is an example of the solution they received in the guided practice. - These problems will be graded by effort, therefore if the lower level student tries the problems they will not be penalized for not understanding it. - If the students are able to complete these three problems it will show you that they understand the information presented in this lesson.