Using Determinants to solve systems of equations

Slides:



Advertisements
Similar presentations
EXAMPLE 3 Use Cramer’s rule for a 2 X 2 system
Advertisements

MATRICES Using matrices to solve Systems of Equations.
Multiplying Decimals Unit 3.5 Pages x 320 = x 800 = 3.3,806 x 10 = 27, ,400 38,060 Warm Up Problems Multiply.
Fractions and Decimals 5.2. Writing a Fraction as a decimal Divide the numerator by the denominator.
Using Inverse Matrices Solving Systems. You can use the inverse of the coefficient matrix to find the solution. 3x + 2y = 7 4x - 5y = 11 Solve the system.
Gabriel Cramer was a Swiss mathematician ( )
3.5 Solution by Determinants. The Determinant of a Matrix The determinant of a matrix A is denoted by |A|. Determinants exist only for square matrices.
Cramer's Rule Gabriel Cramer was a Swiss mathematician ( )
Systems of Linear Equations Let’s say you need to solve the following for x, y, & z: 2x + y – 2z = 10 3x + 2y + 2z = 1 5x + 4y + 3z = 4 Two methods –Gaussian.
4.5 Solving Systems using Matrix Equations and Inverses.
4.4 & 4.5 Notes Remember: Identity Matrices: If the product of two matrices equal the identity matrix then they are inverses.
Inverses and Systems Section Warm – up:
4.5 Solving Systems using Matrix Equations and Inverses OBJ: To solve systems of linear equations using inverse matrices & use systems of linear equations.
4-8 Augmented Matrices and Systems
4.6 Cramer’s Rule Using Determinants to solve systems of equations.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 9 Matrices and Determinants.
4.7 Identity and Inverse Matrices and Solving Systems of Equations Objectives: 1.Determine whether two matrices are inverses. 2.Find the inverse of a 2x2.
Have we ever seen this phenomenon before? Let’s do some quick multiplication…
4.3 Determinants and Cramer’s rule How do you find the determinant of a matrix? How do you find the area of a triangle given 3 sets of coordinates? How.
Algebra a Niagara Falls Get ready for a “Small Quiz” to be written on your grade sheet.
Cramer’s Rule for Matrices You can use properties of matrix determinants for a variety of applications. Today: – Solving 3 variable systems of equations.
3.6 Solving Systems Using Matrices You can use a matrix to represent and solve a system of equations without writing the variables. A matrix is a rectangular.
SLIDE SHOW INSTRUCTIONS To move forward in the presentation press the ‘down arrow’ button (Page down or enter also work) To move backward press the ‘up.
Section 3.2 Solving Equations using Multiplication and Division.
4-8 Cramer’s Rule We can solve a system of linear equations that has a unique solution by using determinants and a pattern called Cramer’s Rule (named.
Notes Over 10.5 Using Cramer’s Rule for a 2 x 2 System
ANSWER is 25% of what number ANSWER 40% 4.What percent of 90 is 36? ANSWER d = 4 ANSWER x = Solve:
Rational Exponents Use the rules for combining fractions when the exponents are rational numbers X 2 3 X 1 2 (()) = X = X = X 7.
2.2 Solving Two- Step Equations. Solving Two Steps Equations 1. Use the Addition or Subtraction Property of Equality to get the term with a variable on.
Warm Up Determine whether each system has zero, one or infinitely many solutions one infinitely many zero 3x + y = 15 3x – 2y = 6 x + 2y = 18.
Determinants Every n  n matrix A is associated with a real number called the determinant of A, written  A . The determinant of a 2  2 matrix.
SYSTEMS OF LINEAR EQUATIONS College Algebra. Graphing and Substitution Solving a system by graphing Types of systems Solving by substitution Applications.
3.8B Solving Systems using Matrix Equations and Inverses.
Notes Over 4.3 Evaluate Determinants of 2 x 2 Matrices
F UNDAMENTALS OF E NGINEERING A NALYSIS Eng. Hassan S. Migdadi Cramer’s Rule for solving linear systems Part 1.
1) Solve. -5t = 60 To get the variable by itself, which number needs to be moved? -5 To move the -5, you have to do the opposite operation. What operation.
Downhill product – Uphill product.
TYPES OF SOLUTIONS SOLVING EQUATIONS
Solve Linear Systems By Multiplying First
Cramer’s Rule (because Cramer RULES!)
Lesson 3.2 Solving Equations with Multiplication and Division
TYPES OF SOLUTIONS SOLVING EQUATIONS
3-3: Cramer’s Rule.
Algebra 2 Chapter 3 Section 3 Cramer’s Rule
Solving Systems Using Matrices
4.3 Determinants & Cramer’s Rule
9-4 Determinants and Cramer’s Rule
9-4 Determinants and Cramer’s Rule
Lesson 13-3: Determinants & Cramer’s Rule
Applying Determinants to solve Systems of Equations 2x2 & 3x3
Cramer’s Rule and Solving Systems of Equations
Objective The student will be able to:
Evaluate Determinants & Apply Cramer’s Rule
Using matrices to solve Systems of Equations
Find the inverse of the matrix
Fundamentals of Engineering Analysis
Use Inverse Matrices to Solve 2 Variable Linear Systems
Students will write a summary explaining how to use Cramer’s rule.
8.1 Matrices and System of Linear Equations
Before: December 4, 2017 Solve each system by substitution. Steps:
Solving Multiplication Equations
4.4 Objectives Day 1: Find the determinants of 2  2 and 3  3 matrices. Day 2: Use Cramer’s rule to solve systems of linear equations. Vocabulary Determinant:
Cramer's Rule Gabriel Cramer was a Swiss mathematician ( )
4.3 Determinants and Cramer’s Rule
TWO STEP EQUATIONS 1. SOLVE FOR X 2. DO THE ADDITION STEP FIRST
Cramer's Rule Gabriel Cramer was a Swiss mathematician ( )
Solve equations using multiplication and division.
Objective The student will be able to:
Lesson 6 Ratio’s and Proportions
Using matrices to solve Systems of Equations
Presentation transcript:

Using Determinants to solve systems of equations 4.6 Cramer’s Rule Using Determinants to solve systems of equations

A system of equations can be written as a matrix 3x + 5y -2x + 7y becomes the matrix x – 6y + 3z 4y – 8z 5x – 3y becomes I will call this type of matrix an operation matrix

Cramer’s Rule using the determinants of two matrices 5x + 4y = 28 Find the determinant of the 3x – 2y = 8 operation matrix

Cramer’s Rule using the determinants of two matrices 5x + 4y = 28 Find the determinant of the 3x – 2y = 8 matrix where one of the variables coefficient are replaced with the answers. When solve for x use Find it determinant We will call this the new answer martix

Cramer’s Rule using the determinants of two matrices Now to solve for x divide the new answer matrix by the operation matrix x is 4; y can be found the same way

Matrix for y New answer matrix Then divide by -22, for the operation matrix

Lets solve this system equations by Cramer’s rule 2x – 3y + z = 5 x + 2y + z = -1 x – 3y + 2z = 1 Need to find the determinants of

Find the determinant We will use this for the denominators in the all the fractions.

Solve for x Replace the x column with the answers. So

Solve for y Replace the y column with the answers. So

Solve for z Replace the z column with the answers. So

Homework Page 192 – 193 # 13,15, 17, 21, 23, 27, 29, 31

Homework Page 192 – 193 # 12, 14, 16, 20, 22, 26, 28, 30