Math 100 - Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014
Systems of Equations
Systems of Equations A system of equations consists of
Systems of Equations A system of equations consists of 1. a collection of unknown variables,
Systems of Equations A system of equations consists of 1. a collection of unknown variables, 2. a collection of equations which relate these unknown variables.
Systems of Equations A system of equations consists of 1. a collection of unknown variables, 2. a collection of equations which relate these unknown variables. A linear equation in two variables x, y is an equation of the form ax + by = c where a, b, c are real-valued constants.
Systems of Equations A system of equations consists of 1. a collection of unknown variables, 2. a collection of equations which relate these unknown variables. A linear equation in two variables x, y is an equation of the form ax + by = c where a, b, c are real-valued constants. A linear equation in three variables x, y, z is an equation of the form ax + by + cz = d where a, b, c, d are real-valued constants.
Systems of Equations A system of equations consists of 1. a collection of unknown variables, 2. a collection of equations which relate these unknown variables. A linear equation in two variables x, y is an equation of the form ax + by = c where a, b, c are real-valued constants. A linear equation in three variables x, y, z is an equation of the form ax + by + cz = d where a, b, c, d are real-valued constants. Today, we will focus on linear systems of equations, which means that each equation in our system of equations is a linear equation.
Solving Systems of Equations Techniques
Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation.
Example Using Substitution A carnival sells a total of 560 tickets made up of adult and children s tickets. Adult tickets cost $4 each, children s tickets cost $2 each. If the total revenue from the sales is $1480, how much of each ticket type was sold?
Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation.
Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation. 2. Elimination: Multiply equations by non-zero constants, then add one equation to a different equation to generate a third equation in one fewer variables than the original two equations.
Example Using Elimination Solve the following system of equations: 5x 3y = 4 (1) 2x + 7y = 1 (2)
Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation. 2. Elimination: Multiply equations by non-zero constants, then add one equation to a different equation to generate a third equation in one fewer variables than the original two equations.
Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation. 2. Elimination: Multiply equations by non-zero constants, then add one equation to a different equation to generate a third equation in one fewer variables than the original two equations. 3. Graphing: The geometric interpretation of a linear equation in n variables is a line in an n-dimensional space. The intersection of all these lines is a solution to the system of equations.
Example Using Graphing Solve the following system of equations: x + y = 3 (3) x y = 4 (4)
Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation. 2. Elimination: Multiply equations by non-zero constants, then add one equation to a different equation to generate a third equation in one fewer variables than the original two equations. 3. Graphing: The geometric interpretation of a linear equation in n variables is a line in an n-dimensional space. The intersection of all these lines is a solution to the system of equations.
Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation. 2. Elimination: Multiply equations by non-zero constants, then add one equation to a different equation to generate a third equation in one fewer variables than the original two equations. 3. Graphing: The geometric interpretation of a linear equation in n variables is a line in an n-dimensional space. The intersection of all these lines is a solution to the system of equations. 4. Matrices: Use row operations to transform an augmented matrix into an upper triangular matrix.
Matrices
Matrices A matrix is a rectangular array of numbers arranged in rows and columns.
Matrices A matrix is a rectangular array of numbers arranged in rows and columns. Example: The following matrix has 2 rows and 2 columns: [ ] 1 2 3 4
Matrices A matrix is a rectangular array of numbers arranged in rows and columns. Example: The following matrix has 2 rows and 2 columns: [ ] 1 2 3 4 We say the shape or size of the matrix is }{{} 2 }{{} 2. # of rows # of columns
Matrices A matrix is a rectangular array of numbers arranged in rows and columns. Example: The following matrix has 2 rows and 2 columns: [ ] 1 2 3 4 We say the shape or size of the matrix is }{{} 2 }{{} 2. # of rows # of columns Example: The following matrix has shape 2 3: [ 1 2 ] 3 4 5 6
Example Using Matrices Solve the following system of equations: 5x 3y = 4 (5) 2x + 7y = 1 (6)
Example Using Matrices Solve the following system of equations: 5x 3y = 4 (5) 2x + 7y = 1 (6) Here we turn this system of equations into an augmented matrix: [ 5 3 ] 4 2 7 1
Example Using Matrices Solve the following system of equations: 5x 3y = 4 (5) 2x + 7y = 1 (6) Here we turn this system of equations into an augmented matrix: [ 5 3 ] 4 2 7 1 Matrix operations which keep a system consistent:
Example Using Matrices Solve the following system of equations: 5x 3y = 4 (5) 2x + 7y = 1 (6) Here we turn this system of equations into an augmented matrix: [ 5 3 ] 4 2 7 1 Matrix operations which keep a system consistent: 1. Multiplying every entry of a row by the same non-zero number.
Example Using Matrices Solve the following system of equations: 5x 3y = 4 (5) 2x + 7y = 1 (6) Here we turn this system of equations into an augmented matrix: [ 5 3 ] 4 2 7 1 Matrix operations which keep a system consistent: 1. Multiplying every entry of a row by the same non-zero number. 2. Adding or subtracting a multiple of one row to another. Example: New row 2 is equal to old row 2 minus 3 times row 1.
Example Using Matrices Solve the following system of equations: 5x 3y = 4 (5) 2x + 7y = 1 (6) Here we turn this system of equations into an augmented matrix: [ 5 3 ] 4 2 7 1 Matrix operations which keep a system consistent: 1. Multiplying every entry of a row by the same non-zero number. 2. Adding or subtracting a multiple of one row to another. Example: New row 2 is equal to old row 2 minus 3 times row 1. Hint: You can often use Operation 1 to make Operation 2 easier, as we will see in this example.
iclicker Question 1 What is the size of the following matrix? 2 3 4 2 2 1
iclicker Question 1 What is the size of the following matrix? 2 3 4 2 2 1 A. 6 1 B. 2 3 C. 3 2 D. 1 6 E. None of the above.
Reducing Augmented Matrices
Reducing Augmented Matrices Reducing a system of equations to an upper triangular matrix allows one to use substitution to solve the system.
Reducing Augmented Matrices Reducing a system of equations to an upper triangular matrix allows one to use substitution to solve the system. Upper triangular augmented matrices are of the form [ 0 where each is a real number. ] and 0 0 0
iclicker Question 2 What is the system of equations corresponding to the augmented matrix below? [ 2 3 ] 4 1 2 3
iclicker Question 2 What is the system of equations corresponding to the augmented matrix below? [ 2 3 ] 4 1 2 3 A. 2x + 3y = 4, x + 2y = 3 B. 3x + 2y = 4, 2x + y = 3 C. 2x + y = 4, 3x + 2y = 3 D. x + y = 4, x + 2y = 4 E. None of the above
Some Additional Thoughts and Questions
Some Additional Thoughts and Questions Some systems of linear equations have exactly 1 solution, some have no solution, and some have infinitely many solutions. These are the only possibilities for linear equations.
Some Additional Thoughts and Questions Some systems of linear equations have exactly 1 solution, some have no solution, and some have infinitely many solutions. These are the only possibilities for linear equations. When a system of linear equations has 1 solution, then this solution can be given by an equation for each variable expressing the exact real number that this variable is equal to. Alternatively, such a solution can be written as an ordered pair.
Some Additional Thoughts and Questions Some systems of linear equations have exactly 1 solution, some have no solution, and some have infinitely many solutions. These are the only possibilities for linear equations. When a system of linear equations has 1 solution, then this solution can be given by an equation for each variable expressing the exact real number that this variable is equal to. Alternatively, such a solution can be written as an ordered pair. How many solutions does the following system have? Why? x + y = 1 (7) 2x + 2y = 2 (8)
Some Additional Thoughts and Questions Some systems of linear equations have exactly 1 solution, some have no solution, and some have infinitely many solutions. These are the only possibilities for linear equations. When a system of linear equations has 1 solution, then this solution can be given by an equation for each variable expressing the exact real number that this variable is equal to. Alternatively, such a solution can be written as an ordered pair. How many solutions does the following system have? Why? x + y = 1 (7) 2x + 2y = 2 (8) What happens when you multiply an equation in your system by zero?
iclicker Question 3 So the following augmented matrix for variables x and y where x corresponds to the first column and y corresponds to the second column. [ 5 3 ] 4 0 2 6
iclicker Question 3 So the following augmented matrix for variables x and y where x corresponds to the first column and y corresponds to the second column. [ 5 3 ] 4 0 2 6 A. x = 1, y = 3 B. x = 1, y = 3 C. x = 1, y = 3 D. x = 1, y = 3 E. None of the above