Learning Target  LT 2: I can model a real-world scenario using a system of equations and find the solution(s).

Modeling with Systems of Equations  There are 150 adults and 225 children at a zoo. If the zoo makes a total of $5100 from the entrance fees, and the cost of an adult and a child to attend is $31, how much does it cost for a parent or child to attend individually. Let a be adults Let c be children How can we improve the definition of these variables? 150a + 225c = 5100 Partner A – What is the meaning of the term 150a? What are its units? Partner B – What is the meaning of the term 225c? What are its units? Describe the meaning of the equation.

Modeling with Systems of Equations  There are 150 adults and 225 children at a zoo. If the zoo makes a total of $5100 from the entrance fees, and the cost of an adult and a child to attend is $31, how much does it cost for a parent or child to attend individually. Let a be the price of an adult ticket Let c be the price of a child’s ticket What is another equation that “a” and “c” must satisfy? 150a + 225c = 5100

Modeling with Systems of Equations 150a + 225c = 5100 a + c = 31 Fold and label your paper: A1) A2) B1) B2) Partner A solves the system of equations using ANY METHOD, explaining their work to Partner B. Partner B listens and asks questions to clarify or understand Partner A.

Modeling with Systems of Equations 150a + 225c = 5100 a + c = 31 Fold and label your paper: A1) A2) B1) B2) Partner B solves the system of equations using a DIFFERENT METHOD, explaining their work to Partner A. Partner A listens and asks questions to clarify or understand Partner B.

Modeling with Systems of Equations 150a + 225c = 5100 a + c = 31 What was effective / ineffective about your solution method? Which method allowed you to solve the problem more easily? Why? Show the solution method that you found more effective on your worksheet and explain why you chose to solve the system of equations in that way.

Modeling with Systems of Equations  Nicole has 15 nickels and dimes. If the value of her coins is $1.20, how many of each type of coin does she have? Let … Partner A – Define the variables. n + d = 15 Partner B – What is the meaning of the first equation that has been written? Write a second equation that the variables must satisfy.

Modeling with Systems of Equations  Nicole has 15 nickels and dimes. If the value of her coins is $1.20, how many of each type of coin does she have? Let n be the number of nickels that Nicole has Let d be the number of dimes that Nicole has What is another way the second equation can be written? n + d = 15

Modeling with Systems of Equations n + d = 15 5n +10d = 120 Fold and label your paper: A1) A2) B1) B2) Partner B solves the system of equations using ANY METHOD, explaining their work to Partner B. Partner A listens and asks questions to clarify or understand Partner B.

Modeling with Systems of Equations n + d = 15 5n +10d = 120 Fold and label your paper: A1) A2) B1) B2) Partner A solves the system of equations using a DIFFERENT METHOD, explaining their work to Partner B. Partner B listens and asks questions to clarify or understand Partner A.

Modeling with Systems of Equations n + d = 15 5n +10d = 120 What was effective / ineffective about your solution method? Which method allowed you to solve the problem more easily? Why? Show the solution method that you found more effective on your worksheet and explain why you chose to solve the system of equations in that way.

Modeling with Systems of Equations

Every group member will solve the problem using a different method (groups of 4 can have one repeated method)

Learning Log Entries Write a summary for today’s Learning Target:  LT 2: I can model a real-world scenario using a system of equations and find the solution(s).

Modeling Mixture Problems How many mL of a 20% acid solution and 12% acid solution should be mixed to yield 300 mL of a 18% solution?

Today’s Learning Target  LT 6: I can multiply 2x2 matrices by hand and larger matrices using technology.

Definition of a Matrix  A matrix is a rectangular arrangement of numbers in horizontal rows and vertical columns. The numbers in a matrix are its elements. 3 columns 2 rows The element in the first row and third column is 5 (a r,c ).

Definition of a Matrix (Cont’d)  The dimensions of a matrix with m rows and n columns is m x n (read “m by n”) 3 columns 2 rows A is a 2x3 matrix.

Definition of a Matrix (Cont’d)  Two matrices are equal if their dimensions are the same and the elements in corresponding positions are equal. BUT…

Quick Check  Identify the dimensions of each matrix.

Quick Check  Identify the position of the circled element of the matrix (a r,c ).

Multiplying Matrices  The product of two matrices A and B is defined only if the number of columns in A is equal to the number of rows in B. If A is a 4 X 3 matrix and B is a 3 X 5 matrix, then the product AB is a 4 X 5 matrix.

4 X 3  3 X 5  4 X 5 M ULTIPLYING T WO M ATRICES 4 rows 5 columns 4 rows 5 columns A  B  AB

Multiplying Matrices  The element in row 1, column 1 of the product of two matrices can be determined by multiplying row 1 by column 1 and adding the products:

Multiplying Matrices  The element in row 1, column 1 of the product of two matrices can be determined by multiplying row 1 by column 1 and adding the products:

Multiplying Matrices  Multiply:

Multiplying Matrices  Multiply:

Multiplying Matrices  Notice: Therefore, matrix multiplication is not commutative.

Multiplying Matrices  Multiply:  Now check your answer by using the calculator.

Multiplying Matrices  Multiply:  Now check your answer by using the calculator.

Multiplying Matrices  Use the calculator to multiply:

The Identity Matrix  The identity matrix is an nxn matrix that has 1’s on the main diagonal and 0’s elsewhere. If A is any nxn matrix and I is the nxn identity matrix, then.

Multiplying Matrices  Check your answers to HW 4.3 (#2 and #4) using the calculator.

Learning Log Entries Write a summary for today’s Learning Target:  LT 6: I can multiply 2x2 matrices by hand and larger matrices using technology.

Today’s Learning Targets  LT 3: I can represent a system of equations using matrices.  LT 4: I can solve a system of equations using an inverse matrix.

Multiplying Matrices (Cont’d)  How can we re-write the following using multiplication ?

Multiplying Matrices (Cont’d)  How can we re-write the following based on the definition of equal matrices.

Systems of Equations  A system of equations of the form: can be re-written as:  What needs to happen in order to solve for x and y?

Inverse Matrices  The inverse of matrix A is A -1 such that.  Find the inverse of A using the calculator and verify that they are inverses.

Inverse Matrices  The inverse of matrix A is A -1 such that.  Find the inverse of A using the calculator and verify that they are inverses.

Systems of Equations Ex) Solve for the variable matrix using a calculator, showing your work.

Systems of Equations Ex) Solve for the variable matrix using a calculator, showing your work.

Learning Log Entries Write a summary for today’s Learning Target:  LT 3: I can represent a system of equations using matrices.

Today’s Learning Target  LT 5: I can write the inverse matrix A -1 for a 2x2 matrix A and describe their product.

The Determinant Each square matrix (nxn) is associated with a real number called its determinant. The determinant of matrix A is denoted by det A or.

The Determinant a) Find det A if b) Evaluate

Inverse Matrices The inverse of the matrix is, provided.

Inverse Matrices a) Find A -1 if b) Find the inverse of

Learning Log Entries Write a summary for today’s Learning Target:  LT 5: I can write the inverse matrix A -1 for a 2x2 matrix A and describe their product.

Today’s Learning Targets  LT 4: I can solve a system of equations using an inverse matrix.

Solving a Matrix Equation Solve the matrix equation for x and y by hand.

Solving a Matrix Equation Solve the matrix equation for x and y by hand.

Solving a Matrix Equation Solve the matrix equation for x and y by hand.

Systems of Equations Ex) Write the system of linear equations as a matrix equation and solve using the inverse matrix.

Systems of Equations Ex) Write the system of linear equations as a matrix equation and solve using the inverse matrix.