WEEK 6 Day 2

Progress report Thursday the 11th.

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.

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

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.

Method 1 Graphing (plot) page 213

When the two lines intersect, the system of equations is called independent and consistent.

Page 213When the two lines are parallel, the system of

equations is called inconsistent.

Page 213When the two lines coincide, the system of

equations is called dependent.

Method 1 Graphing (plot) page 213

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

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.

Addition Subtraction

2x + 3y = -4X – 2y = 5

2 (X – 2y = 5) 22x – 4y = 10

Only 1 equation but both sides.

2x + 3y = -4- 2x – 4y = 10

7y = - 14

7 7 Y = -2

No “x”.

Substitute y = -2 in either original equation.

2 x + 3(-2) = -4 2x + -6 = -4 2x = -4 + 6 2 x = 2 x = 1

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.

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”.

Check by substituting

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.

Page 215

3x + y = 3

2x - 4y = 16Solve for x or y. (y) 3x + y = 3

y = -3x + 3

Page 215

2x - 4y = 16 2x – 4(-3x + 3) = 162x + 12x - 12 = 16

14x = 28 x = 2

No “y”.

Page 217

A special case of the substitution method is the comparison method:

a = cb = ca = b

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.

Page 218Section 6.1

6.2 OTHER SYSTEMS OF EQUATIONS

A literal equation is one in which letter coefficients are used in place of numerical coefficients.

No numbers.

In the equations, a and b represent known quantities or coefficients,

and x and y are the variables or unknown quantities.

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.

6.2 OTHER SYSTEMS OF EQUATIONS page 222

(a) ax + by = ab (a) x + aby = b(b) bx – ay = (b) x - aby =

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 +

6.2 OTHER SYSTEMS OF EQUATIONS page 222

Factor the equation.x + x = b +

x (+ ) = b (+ )

6.2 OTHER SYSTEMS OF EQUATIONS page 222

Factor the equation.x + x = b +

x (+ ) = b (+ )(+ ) (+ )

x = b

6.2 OTHER SYSTEMS OF EQUATIONS page 222

Substitute b for x.

ax + by = ab bx – ay =

a(b) + by = ab

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)

End Week 6 Day 1

6.2The equations in the system are not linear, or first-degree, equations.

The Lowest Common Denominator is: xy

6y + 4x = -2xy

6.3 SOLVING A SYSTEM OFTHREE LINEAR EQUATIONSPage 224

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.

Graphical solutions of three linear equations with three unknowns are not used becausethree-dimensional graphing is required and is not practical by hand.

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.

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)

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)

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,

6.3 SOLVING A SYSTEM OF THREE LINEAR EQUATIONS page 226

Substitute 2 for z in one of the equations and solve for y.

Review

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.

Product Factoring

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.

5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195

3 x 20

5 a b 6 yReorganize like terms.

3 20 x 5 6 a b y

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

5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195

Page 196

In class exercise week 6 day 2.