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I. Systems of Linear Equations in Two Variables
A. Terminology
1. All variables in a linear equation have an exponent of _____.
2. The solution to a system of linear equations in two variables is the set of all ordered pairs for which __________ equations are true.
3. If a system has at least one solution it is said to be ___________________.
4. If a system does not have a solution it is said to be ___________________.
5. If a system has an infinite number of solutions, the equations are said to be __________________.
B. Substitution Method
1. Procedure
a. In either equation solve for one _______________ in terms of the other variable.
b. _________________ the expression from step 1 into the other equation.
c. ____________ the equation in one variable
d. _________________ the value found in step c into one of the original equations to find the value of the other ________________.
2. Solve {█(x+4y=5@2x-y=-8)}
a. Solve for x in the first equation: x = ____________
b. Substitute this expression for x in the other equation: 2(5-4y) – y = -8
c. Solve for y. y = ____
d. Substitute 2 into the first equation for y: x + 4(2) = 5 and solve for x. x = ____
C. Elimination Method
1. Procedure
a. Multiply one or both equations by the appropriate number so that the sum of the ______________________ of one variable is _______.
b. Add these two equations to obtain an equation in ________ variable.
c. . ____________ the equation in one variable
d. _________________ the value found in step c into one of the original equations to find the value of the other ________________.
2. Solve {█(3x-2y=20@2x+6y=-38)}
a. Multiply the first equation by 3: {█(9x-6y=60@2x+6y=-38)}
b. Add these two equations: 11x = _____
c. Solve for x: x = ______
d. Substitute this value into one of the original equations and solve for y. y = _____
D. Special Cases
1. If we eliminate both variables, leaving us with a false statement such as 0 = 5, there is ______ solution to the system. This is an ___________________ system.
2. If we eliminate both variables, leaving us with a true statement such as 0 = 0, there are __________________ many solutions. This is a _________________ system.
3. Solve {█(3x-12y=6@-2x+8y=-4)}
a. Multiply the 1st equation by ____ and the 2nd equation by _____ to produce{█(6x-24y=12@-6x+24y=-12)}
b. Adding these two equations gives us 0 = 0.
c. There are infinitely many solutions. To represent those solutions we solve for one of the variables in either equation. Solving for x in the first equation gives x = 4y + 2
d. For any value you choose for y, x will have to be 4y + 2. Therefore we can represent our solutions as (4y + 2, y).
II. Systems of Linear Equations in Three Variables
A. Gaussian Elimination
1. Our goal is to reduce the system into an equivalent system that is in _____________ form and then use ___________________________ to solve for each variable. This method is referred to as the _____________________________ method.
2. The algebraic operations we use to produce an equivalent system are called elementary ________ operations. These are:
a. ____________________ any two equations
b. ________________ an equation by a nonzero constant.
c. Add a __________________ of one equation to another equation.
3. Solve the system using Gaussian elimination
{█(x+2y+3z=11@3x+8y+5z=27@-x+y+2z=2)}
a. Multiply the first equation by ____ and add to the second equation.[-3R1 + R2 = new R2]
-3x – 6y – 9z = -33
3x + 8y + 5z = 27
2y – 4z = -6 new equation two
b. This gives us the equivalent system: {█(x+2y+3z=11@ 2y- 4z = -6 @-x+y+2z=2)}
c. Next add the 1st equation to the ________ equation. [R1 + R3=new R3]
This gives us the system: {█(x+2y+3z=11@2y- 4z = -6@ 3y+5z=13)}
d. Multiply the 2nd equation by ½ to get a coefficient of 1 for y {█(x+2y+3z=11@y- 2z = -3@ 3y+5z=13)}
e. Multiply the 2nd equation by ______ and add to the 3rd equation
{█(x+2y+3z=11@y- 2z = -3@ 11z=22)}
f. This gives us triangular form
g. Solving the 3rd equation for z, gives z = ______
h. Substituting this value for z in equation 2 and solving for y gives y = _____
i. Now substitute the y and z values into equation 1 and solve for x. x = _____.
B. Augmented Matrix
1. Instead of writing the 3 equations in each step, we only write the coefficients in each step. We write the coefficients in a rectangular array called and augmented ________.
2. Write the system above as an augmented matrix: [█(1 2 3 11@3 8 5 27@-1 1 2 2)]
a. Multiply Row 1 by ____ and add to Row 2 and replace Row 2 [█(1 2 3 11@0 2 -4 -6@-1 1 2 2)]
b. Add Row ____ to Row 3 and replace Row 3 [█(1 2 3 11@0 2 -4 -6@0 3 5 13)]
c. Multiply Row ____ by ½ [█(1 2 3 11@0 1 -2 -3@0 3 5 13)]
d. Multiply Row 2 by ____ and add to Row 3 and replace Row 3 [█(1 2 3 11@0 1 -2 -3@0 0 11 22)]
e. This matrix is in triangular form and corresponds to the system {█(x+2y+3z=11@y-2z=-3@11z=22)}
f. Next we use back substitution as we did above to solve for x, y and z
C. Row-Echelon Form
1. Triangular form only requires zeros below the ____________.
2. Row-Echelon form is triangular form with ____’s down the diagonal
a. To put the matrix above in row-echelon form, Multiply Row ____ in the final matrix by 1/11.
3. Reduced row-echelon form requires _______ below and above the diagonal and _____ down the diagonal.
a. The process of reducing system into reduced row-echelon form is called _________________________ elimination.
D. Determine the quadratic function whose graph passes though the three points (1, -2), (2, 5), and (5, 50)
1. A quadratic function has the form f(x) = ax2 + bx + c.
2. If the point (1, -2) lies on the graph it satisfies the equation y = ax2 + bx + c.
a. Substitute ____ for x and ____ for y to get -2 = a + b + c
3. If the point (2, 5) lies on the graph, we get 5 = ____a + ____b + c
4. If the point (5, 50) lies on the graph, we get ____ = 25a + 5b + c
5. This gives us the system {█(-2=a+b+c@5=4a+2b+c@50=25a+5b+c)}.
6. The solution to the system is the ordered triplet (_____________). Therefore the function is f(x) = 2×2 + x – 5.
III. Inconsistent and Dependent Systems in Three Variables
A. If the bottom row of the augmented matrix turns out to be [0 0 0 1], this corresponds to the equation 0x + 0y + 0z = _____. This is an untrue statement and therefore this is a(n) __________________ system. It will have _________________ solution(s).
B. If the bottom row of the augmented matrix turns out to be [0 0 0 0], this corresponds to the equation 0x + 0y + 0z = _____. This is a true statement and therefore this is a(n) __________________ system. It will have ________________ solution(s).
C. If we have a dependent system, the infinite solutions can be described in terms of one variable.
1. One row of the matrix will be true for all values of the variables.
2. Use the other two equations to solve for two of the variable in terms of the third.
3. Solve the system {█(x+4y+2z=5@3x+7y+z=0@2x+5y+z=1)}
a. Converting the augmented matrix to row-echelon form gives [█(1 4 2 5@0 1 1 3@0 0 0 0)]
b. Since the 3rd row corresponds to an equation which is always true, we use the first two rows to solve for two of the variables in terms of the third. The first two rows correspond to the equations x + 4y + 2z = 5 and y + ____ = _____
c. Solving the second equation for y gives y = 3 – z.
d. Substitute this expression for y in the first equation to get x + 4(3 – z) + 2z = ____
e. Solve this equation for x to get x = ___________
f. The infinite solutions can be given in terms of z: (2z-7, 3-z, z)
III. Matrix Operations
A. Definitions
1. If a matrix has 3 rows and 2 columns, the size of the matrix is __________.
2. If the number of rows equals the number of columns, it is called a ____________ matrix.
3. If a matrix only has one row, it is called a _____________ matrix.
4. If a matrix only has one column, it is called a _______________ matrix.
5. An _______________ matrix is a square matrix with 1’s down the diagonal (from left to right) and zeros everywhere else.
B. Adding and Subtracting Matrices
1. To add or subtract matrices, they must be the same __________.
2. To add or subtract matrices, we simply add or subtract the corresponding entries.
3. Find x, y and z: [■(3&0&-1@1&2&3)]-[■(1&-5&-3@4&-5&0)]=[■(x&y&z@4&-5&0)]. x = ____, y = ____, and z = _____.
C. Scalar Multiplication.
1. We can multiply any matrix by a real number. This real number is called a __________.
2. To perform scalar multiplication we multiply every entry in the matrix by the scalar.
3. Find a and b: 3[■(2&-4@0&5)]=[■(a&b@0&15)]. a = _____, and b = _____.
D. Matrix Multiplication
1. Two matrices can only be multiplied together if the number of ______________ of the first matrix is equal to the number of ______________ of the second matrix.
2. To multiply matrices we multiply the elements of a row in the first matrix with the corresponding elements of a column in the second matrix. For example, the element in the 2nd row and 3rd column of matrix AB is the sum of the products of the elements in row _____ of A and the corresponding elements in column _____ of B.
3. Find p and q: [■(1&3&1@-1&0&4)][■(4&-1@1&0@-2&5)]=[■(p&q@-12&21)].
p = 1(___) + 3(1) + (____)(-2) = _____
q = (___)(-1) + 3(0) + 1(___) = _____
4. Identity Property for Matrix Multiplication
a. For any m x n matrix A, ImA = ____ and AIn = _____. If A is an n x n square matrix, then AIn = InA = ____.
5. Find a, b, c, and d: [■(3&-1@2&5)][■(1&0@0&1)]=[■(a&b@c&d)]. a = ____, b = ____, c = ____, d = ____.
V. Inverse Matrices
A. If A-1 is the multiplicative inverse of a square matrix A, then A A-1 = A-1A = ____.
B. Not every square matrix has an inverse.
1. If a square matrix has an inverse we say it is ________________.
2. If a square matrix does not have an inverse, we say it is _________________.
C. Every square matrix has a number associated with it called a determinant.
1. The determinant of the 2 x 2 matrix A=[■(a&b@c&d)], denoted by |A|, is _______ – _______.
2. Given matrix B=[■(3&1@2&4)], |B| = ______.
D. The Inverse of a 2 x 2 matrix.
1. If A = [■(a&b@c&d)], and |A| ≠ 0, then A-1 = 1/|A| [■(d&-b@-c&a)].
2. Given A=[■(4&11@2&6)], then A-1 = [■(3&(-11)⁄2@x&y)]. x = _____ and y = _____.
E. Find the Inverse of a 3 x 3 Matrix using a graphing calculator
1. You should watch the Graphing Calculator Tutorials by David Williamson by going to the URL: http://dtc.pima.edu/~dwilliamson/TI/indexti.html)
a. Watch the topics “Matrix-Dimension and Editing,” “Matrix-Determinate and Inverse” and “Matrix Equation
2. Find the inverse of [■(2&4&1@-1&1&-1@1&4&0)] using a graphing calculator
a. Select [2nd][MATRIX], then use the right arrow to move to the EDIT menu. Press [ENTER] to select [A].
b. Enter the size as a _____ x _____, then type each entry of the given matrix moving left to right across the 1st row, then 2nd row and then 3rd row. (Press ENTER after each number in the matrix.)
c. Return to the home screen by selecting [2nd][QUIT].
d. Place Matrix [A] on the home screen by selecting [2nd][MATRIX]. Using the NAMES menu, select [A]. This will place [A] on the home screen.
e. To get the inverse press [x-1], then [ENTER]. A-1 will appear on the screen.
3. You will be responsible for knowing the algebraic technique of finding the inverse of a 2 x 2 matrix given in D above, but may use a calculator to find inverses of 3 x 3 matrices, 4 x 4 matrices, or higher.
VI. Solving Systems of Equations Using an Inverse Matrix
A. A system of equations can be written as the matrix equation AX = B. A is called the ____________________ matrix. X is called the ____________________ matrix. B is called the ____________________ matrix.
B. If A is invertible, the X = A-1(_____).
C. Solve the system {█(3x+2y-z=-4@x+3y+2z=1@-2x+z=1)} using an inverse matrix. View the Graphing Calculator Tutorial, “Matrix Equations.”
1. Let A = [■(3&2&-1@1&3&2@-2&0&1)], X = [■(x@y@z)], and B = [■(-4@1@1)]
2. X = A-1B. Use a calculator to find A-1B. This gives x = ____, y = ____, and z = ____.
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