Math Studio College Algebra
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1 Math Studio College Algebra Rekha Natarajan Kansas State University November 19, 2014
2 Systems of Equations
3 Systems of Equations A system of equations consists of
4 Systems of Equations A system of equations consists of 1. a collection of unknown variables,
5 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.
6 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.
7 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.
8 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.
9 Solving Systems of Equations Techniques
10 Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation.
11 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?
12 Solving Systems of Equations Techniques 1. Substitution: Solve for one variable in one equation. Substitute this value back into the other equation.
13 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.
14 Example Using Elimination Solve the following system of equations: 5x 3y = 4 (1) 2x + 7y = 1 (2)
15 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.
16 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.
17 Example Using Graphing Solve the following system of equations: x + y = 3 (3) x y = 4 (4)
18 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.
19 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.
20 Matrices
21 Matrices A matrix is a rectangular array of numbers arranged in rows and columns.
22 Matrices A matrix is a rectangular array of numbers arranged in rows and columns. Example: The following matrix has 2 rows and 2 columns: [ ]
23 Matrices A matrix is a rectangular array of numbers arranged in rows and columns. Example: The following matrix has 2 rows and 2 columns: [ ] We say the shape or size of the matrix is }{{} 2 }{{} 2. # of rows # of columns
24 Matrices A matrix is a rectangular array of numbers arranged in rows and columns. Example: The following matrix has 2 rows and 2 columns: [ ] 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 ]
25 Example Using Matrices Solve the following system of equations: 5x 3y = 4 (5) 2x + 7y = 1 (6)
26 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 ]
27 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 ] Matrix operations which keep a system consistent:
28 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 ] Matrix operations which keep a system consistent: 1. Multiplying every entry of a row by the same non-zero number.
29 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 ] 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.
30 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 ] 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.
31 iclicker Question 1 What is the size of the following matrix?
32 iclicker Question 1 What is the size of the following matrix? A. 6 1 B. 2 3 C. 3 2 D. 1 6 E. None of the above.
33 Reducing Augmented Matrices
34 Reducing Augmented Matrices Reducing a system of equations to an upper triangular matrix allows one to use substitution to solve the system.
35 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
36 iclicker Question 2 What is the system of equations corresponding to the augmented matrix below? [ 2 3 ]
37 iclicker Question 2 What is the system of equations corresponding to the augmented matrix below? [ 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
38 Some Additional Thoughts and Questions
39 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.
40 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.
41 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)
42 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?
43 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 ]
44 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 ] 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
3. Replace any row by the sum of that row and a constant multiple of any other row.
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