week 6 day 2. progress report thursday the 11 th
TRANSCRIPT
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WEEK 6 Day 2
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Progress report Thursday the 11th.
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Objectives Solve systems of equations by substitution. Solve systems of equations by the addition-subtraction method. Evaluate determinants using determinant properties. Use Cramer’s rule. Use the method of partial fractions to rewrite rational expressions as the sum or the differenceof simpler expressions.
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6.1 SOLVING A SYSTEM OF TWO LINEAR EQUATIONS page212
In this section we shall study solutions by:1. Graphing2. Addition-subtraction method3. Method of substitution
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6.1 SOLVING A SYSTEM OF TWO LINEAR EQUATIONS page213
Any ordered pair (x, y) that satisfies both equations is called a solution, or root, of the system.
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Method 1 Graphing (plot) page 213
When the two lines intersect, the system of equations is called independent and consistent.
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Page 213When the two lines are parallel, the system of
equations is called inconsistent.
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Page 213When the two lines coincide, the system of
equations is called dependent.
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Method 1 Graphing (plot) page 213
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Method 2 Add and Subtract page 214
The first algebraic method (second method over all) is called the:
addition-subtraction method. (sometimes called the elimination method)eliminating X or Y
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Page 214multiply each side of one or both equations by some
number so that the numerical coefficients of one of the variables are of equal absolute value.
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Addition Subtraction
2x + 3y = -4X – 2y = 5
2 (X – 2y = 5) 22x – 4y = 10
Only 1 equation but both sides.
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2x + 3y = -4- 2x – 4y = 10
7y = - 14
7 7 Y = -2
No “x”.
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Substitute y = -2 in either original equation.
2 x + 3(-2) = -4 2x + -6 = -4 2x = -4 + 6 2 x = 2 x = 1
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multiply each side of one or both equations by some number
2x + 3y = -4X – 2y = 5
2 2x + 3y = -4 24x + 6y = -8
4 (X – 2y = 5) 44x – 8y = 20
Both equations.
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multiply each side of one or both equations by some number
4x + 6y = -8- (4x – 8y) = 20
14y = - 28 14 14 y = -2
No “x”.
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Check by substituting
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Method 3 Substitution Page 215
The second algebraic method (3rd method over all) of solving systems of linear equations is called the method of substitution.
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Page 215
3x + y = 3
2x - 4y = 16Solve for x or y. (y) 3x + y = 3
y = -3x + 3
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Page 215
2x - 4y = 16 2x – 4(-3x + 3) = 162x + 12x - 12 = 16
14x = 28 x = 2
No “y”.
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Page 217
A special case of the substitution method is the comparison method:
a = cb = ca = b
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Page 217
Comparison method:3x – 4 = 5y6 – 2x = 5y
Since the left side of each equation equals the same quantity, we have:3x – 4 = 6 – 2x
This eliminates the variable y.
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Page 218Section 6.1
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6.2 OTHER SYSTEMS OF EQUATIONS
A literal equation is one in which letter coefficients are used in place of numerical coefficients.
No numbers.
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In the equations, a and b represent known quantities or coefficients,
and x and y are the variables or unknown quantities.
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6.2 OTHER SYSTEMS OF EQUATIONS page 222
ax + by = ab Multiply by abx – ay = Multiply by b
(a) ax + by = ab (a) Why a, b ?(b) bx – ay = (b) Y is then out.
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6.2 OTHER SYSTEMS OF EQUATIONS page 222
(a) ax + by = ab (a) x + aby = b(b) bx – ay = (b) x - aby =
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6.2 OTHER SYSTEMS OF EQUATIONS page 222
Add the two equations. x + aby = b + (x – aby) =
Why add?
Y is then out. x + x = b +
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6.2 OTHER SYSTEMS OF EQUATIONS page 222
Factor the equation.x + x = b +
x (+ ) = b (+ )
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6.2 OTHER SYSTEMS OF EQUATIONS page 222
Factor the equation.x + x = b +
x (+ ) = b (+ )(+ ) (+ )
x = b
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6.2 OTHER SYSTEMS OF EQUATIONS page 222
Substitute b for x.
ax + by = ab bx – ay =
a(b) + by = ab
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6.2 OTHER SYSTEMS OF EQUATIONS page 222
Substitute b for x. ax + by = ab
a(b) + by = ab - a(b) = - ab
by = 0x = b and y = 0 (b, y)
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End Week 6 Day 1
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6.2The equations in the system are not linear, or first-degree, equations.
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The Lowest Common Denominator is: xy
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6y + 4x = -2xy
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6.3 SOLVING A SYSTEM OFTHREE LINEAR EQUATIONSPage 224
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6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 224
The graph of a linear equation with three variables in the form is a plane.
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Graphical solutions of three linear equations with three unknowns are not used becausethree-dimensional graphing is required and is not practical by hand.
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6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 225
Let’s choose to eliminate x first. To eliminate x from any pair of equations, such as (1) and (2), multiply each side of Equation by your chosen number and subtract.
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6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 225
Choosing to multiply each side of Equation (1) by 2 you get.
(2) (2)
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6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 225
To eliminate x from any other pair of equations, such as (1) and (3), multiply each side ofEquation (1) by 3 and add.
(3)(3)
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6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 226
We have now reduced the system of three equations in three variables to a system of twoequations in two variables,
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6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 226
Substitute 2 for z in one of the equations and solve for y.
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Review
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A product is the result obtained by multiplying two or more quantities together.
Factoring is finding the numbers or expressions that multiply together to make a given number or equation.
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Product Factoring
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5.4 EQUIVALENT FRACTIONS page 189
Two fractions are equivalent when both the numerator and the denominator of onefraction can be multiplied or divided by the same nonzero number.
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5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195
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3 x 20
5 a b 6 yReorganize like terms.
3 20 x 5 6 a b y
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3 20 x 5 6 a b y
Factor each of the terms in the numerator and denominator.
Divide by common factors.
60 a y30 b x
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5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195
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Page 196
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In class exercise week 6 day 2.