Notes #3-___
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7.1 Solving Systems of Two Equations (568)
The solution to a system of linear equations is the ordered pair (x, y) where the lines intersect! A solution can be substituted into both equations to make true statements.
Consistent – solution(s) Inconsistent – Ø
Dependent (∞) Independent (1 or Ø)
1. Solve an equation for one of the variables (if necessary).
2. Substitute the expression for the variable and solve.
3. Substitute into either equation to get the other value.
y = 2x + 31. Not necessary.
3x + 2y = -12.3x + 2(2x + 3) = -1
3x + 4x + 6 = -1
7x + 6 = -1,7x = -7, x = -1
3.y = 2(-1) + 3 = -2 + 3 = 1Ans: (-1, 1)
Ex.1Solve using substitution:
a)3x + 2y = 14b)-3x + y = 7
x – 2y = 10-6x + 2y = -8
You do not have to
substitute for a single variable (you can use an expression).
Graphing
Linear Combination
or Elimination
Ex.2Solving nonlinear systems:
a)y = x2 – 1b)x + y = 6
x2 + y2 = 13
Ex.3Graph and determine the point of intersection.:
a)3x + y = 4b)3x + 2y = 6
x – 2y = 6y = ln(x – 1)
1.Arrange like terms in columns (if necessary).
2.Multiply so that one set of variables are opposites.
3.Add and then solve.
4.Substitute into either equation to get the other value.
y = 2x + 31. -2x both sides-2x + y = 3
3x + 2y = -13x + 2y = -1
2.Eliminate y if we multiply eq.#1 by -2
-2(-2x + y = 3) = 4x – 2y = -6
3x + 2y = -1
3. 7x = -7
x = -1
4.y = 2(-1) + 3 =
-2 + 3 = 1 Ans: (-1, 1)
Ex.4Solve using elimination:
a)Find the product of the x & y coordinates of the
solution: 3x – 2y = 4
2x + 5y = -2
b)2x – 4y = 8
x – 2y = 4
Ex.5Three gallons of a mixture is 60% water by volume.
Determine the number of gallons of water that must be
added to bring the mixture to 75% water.
Ex.6A car radiator contains 10 quarts of a 30% antifreeze
solution. How many quarts will have to be replaced
with pure antifreeze if the resulting solution is to be
50% antifreeze?
Summary:
Ex.7A machine takes 3 minutes to form a bowl and 2
minutes to form a plate. The material costs $0.25 for
a bowl and $0.20 for a plate.How many bowls and
plates were made if the machine ran for 8 hours and
$44 was spent on materials?
Ex.8A company determines its supply and demand
equations for a product to be:S: p = x2 + 4x + 40
D: p = 50 – 3x
Find the equilibrium point in terms of x (thousands of
the item) and p (the price). What does it mean?
What relationship does the company want to have
between the supply and the demand?
Notes #3-___
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7.2 Matrix Algebra Day 1 (579-583)
Order (dimensions) 3 × 2
(rows) r × c (columns)
A =
If r = c then it is a square matrix
entry/element
a32 = -2 (row 3, column 2)
Ex.1Determine the order:
Is it a square matrix?
Identify:B =
b33 =b24 =
In addition & subtraction, matrices must have the same order.
Add/subtract the corresponding elements. You multiply all elements of a matrix by a scalar multiple (think distributive).
Ex.2=
Identities:
Addition:3 + 0 = 3
Multiplication:3 · 1 = 3
Matrix Addition:
Matrix Multiplication (not commutative AB ≠ BA):
Ex.3Find the product AB:
Ex.4Find the product:
Ex.5Redo Ex.3 & Ex.4 on the graphing calculator. List the
steps in your notes.
Identity:Inverse:
3 + 0 = 33 + (-3) = 0 (opposite)
3 · 1 = 33 · = 1 (reciprocal)
Ex.6Verify:
Determinant: If A = , then det(A) = = ad – bc
Ex.7Find the value of:
a)b)det
Summary:
Ex.8Find the inverse of a 2 × 2:
A =
1. Find det(A), no A-1 if det(A) = 0.
2.Exchange a & d.
3.Opposite signs on b & c.
4.Multiply by the reciprocal of det(A).
Ex.9Find the inverse of:
Notes #3-___
Date:______
7.2 Matrix Algebra Day 2 (584-589)
A matrix that has a nonzero determinant is called a
nonsingular matrix (which means it has an inverse).
The minor (short for minor determinant) of an element of an
nth order determinant can be found by deleting the row
and column containing the element.
A = The minor of a11 is .
The cofactor of an element is (-1)i+j Mij. So the cofactor of
a11 is (-1)1+1(-12 – 10) = (-1)2(-22) = 1(-22) = -22.
Another way to deal with the signs
is to follow the pattern that a11 is +
and adjacent elements (horizontal
& vertical) are the opposite sign.
To evaluate a determinant of the nth order, multiple the elements of a row or column by their corresponding cofactors:
-2(+) + 0(-) + (-6)(+)=
Ex.1Find the value of:
a)b)det
Lattice method:
Ex.2Find the inverse of a 2 × 2:
A =
Adjoin the identity and perform row
opsto get the identity on the left:
If you get the 0s first you will have less fractions to
deal with.
Row operations are basically linear combinations.
Ex.3Find the inverse:
Cramer’s Rule:
If you get you have coincident lines (planes) & is Ø.
Ex.4Use Cramer’s Rule to solve:5x – 3y = 4
2x + 4y = 1
Ex.5Use Cramer’s Rule to solve for z:
a) x – 2y – z = 4b) x – y + z = 8
4x + y + z = 72x + 3y – z = -2
x + 3y – 4z = -13x – 2y – 9z = 9
S R1 & R2
-2 R1 + R3
R3 – 2R2
-5R2 + R3
-.5 R3
-R3 + R1
R2 + R1
A-1 =
Notes #3-___
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7.3Multivariate Linear Systems & Row Operations (594)
Triangular Form (leading coefficients = 1):
Ex.1Use back substitution: 1x – 2y – 2z = 7
1y – 2z = -7
1z = 3
Gaussian Elimination:
Interchange rows or use
elimination to change rows.
Ex.22x + y – z = 0
x – y + 2z = -3
-x + 2y – 3z = 7
x – y + z = 8
2x + 3y – z = -2
3x – 2y – 9z = 9
Row Echelon Form (REF):
Ex.33x – 6y + 3z = 7
2x + y + 4z = 2
x – 2y + z = 4
A is called the
coefficient matrix.
Gauss-Jordan Elimination (Reduced REF):
Ex.43x + 2y + z = 5Ex.52x + y – z = 1
x – y + z = 43x + 3y + 2z = 4
x + 3y – 2z = -3x – y – z = 0
Ex.6x + y – 3z = -1Ex.7Nonsquare Systems:
y – z = 0x – 2y – z = -5
-x + 2y = 12x + y + z = 5
Inverse matrices can be used to solve systems:
2x = 4 is solved by using inverses: , x = 2
Ex.7
A-1 · A · X = A-1 · B
Summary:
Ex.8Solve using inverse matrices:
a)x + 2y + z = -4b) x + y + z + w = 4
2x – y + z = -42x – y + z = 0
x + 3y – z = -73x + 2y + z – w = 6
x – 2y – 2z + 2w = -1
Ex.9You borrow $10,000; some at 18%, 15% and 9%.
You borrow 3 times as much at 15% as you borrow at
18% and the interest is $1244.25. How much each?
Ex.10Determine the quadratic function that contains the
points: (1, 3), (2, 10) & (-2, -6).
Notes #3-___
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Not unique:
Unique?
7.4Partial Fraction Decomposition (608)
I.Intro:
Ex.1
How can we get back?
Ex.2 = =
PFD: writing rational expressions as the sum of simpler ones.
II.Distinct (no repeated) linear factors in the denominator.
Step #1: Factor the denominator.
Step #2: Write the factors as a sum.
Ex.3 ==
Step #3:How do we solve rational equations?
Multiply by the LCD.
Step #4:Solve the resulting system of equations.
Strategic values of x
may help you solve the system. See Ex.3.
Ex.4
III.Repeated Linear Factors
Ex.5
IV.Irreducible Distinct Quadratic Factors
Ex.6
Summary:
V.Irreducible Repeated Quadratic Factors
Ex.7
Notes #3-___
Date:______
7.4Partial Fractions Day 2 (613)
I.Intro:
Ex.1Write the terms for the PFD (don’t solve):
=
Ex.2Decompose:
Ex.3Use division to rewrite in the form :
Where is the low battery warning?
Summary:
II.Improper Fractions (Degree of N > D):
Ex.4Use division to rewrite in the form and
then find the PFD of : f(x) =
Compare the rational f(x) graph to the terms of the PFD.
Ex.5Use division to rewrite in the form and
then find the PFD of :
Notes #3-___
Date:______
7.5 Systems of Inequalities in Two Variables (617)
The graph of a linear inequality is a half-plane.
More lead? / Number Line / Coordinate PlaneOpen dot / Dotted line
& ≥ / Closed (solid) dot / Solid Line
Use a test point for each inequality.
Ex.1Graph (are they bounded or unbounded):
a)x < 4, y ≥ -3b)x ≥ 1, x < 4,y ≥ -1
and x – y ≥ 4& y ≤ x2 + 1.
Ex.2Write a system of
inequalities:
Ex.3Solve the system of inequalities.
a)b)
y -x2 – 2x + 2x2 + y2 8
Linear Programming:
Constraints: a system of inequalities
Feasible Region: solution set
Objective Function: an equation applied to a feasible region
Optimal Solution: maximum or minimum of the solution set
Fundamental Theorem of Linear Programming: the
maximum or minimum always occurs at a vertex.
Ex.4Mr. Jones is bakingas many cakes & pies as possible
using the 18 eggs 15 cups ofmilk he has on hand.
His cake recipe requires 2 eggs and 1 cup of milk and
his pie recipe requires 1 egg and 1.5 cups of milk.
a)Find the constraints and
graph the feasible region.
b)Write the objective function
and determine how many
cakes pies should be made.
Summary:
Ex.5It takes 2 hrs to manufacture a pair of skis and 1 hr
for a snowboard. The finishing time for bothis 1 hr.
The maximum time available for manufacturing is
40 hrs and for finishing is 32 hours eachweek. The
profit for a pair of skis is $70 the profit for a
snowboard is $50. The manufacturermust produce
at least 8 snowboards every week because of a
contract with a sporting goods store.
a)Find the constraints and graph the feasible region.
b)Write the objective function
find the maximum profit.