WEEK 6 Day 1 Progress report Thursday the 11 th. Average score 87.6% In class exercise Week 5 Day 1 5.4 EQUIVALENT FRACTIONS page 192 A fraction is in lowest terms when its numerator and denominator have no common factors except 1. 9 and a squared a can be reduced. 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. ProductFactoring 5.4 EQUIVALENT FRACTIONS page 189 Two fractions are equivalent when both the numerator and the denominator of one fraction can be multiplied or divided by the same nonzero number. 5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195 Reorganize like terms. Factor each of the terms in the numerator and denominator. Divide by common factors. 60 a y 30 b x 5.5 MULTIPLICATION AND DIVISION OF ALGEBRAIC FRACTIONS page 195 Page 196 Objectives Solve systems of equations by substitution. Solve systems of equations by the addition- subtraction method. Evaluate determinants using determinant properties. Use Cramers rule. Use the method of partial fractions to rewrite rational expressions as the sum or the difference of simpler expressions. 6.1 SOLVING A SYSTEM OF TWO LINEAR EQUATIONS page212 In this section we shall study solutions by: 1. Graphing 2. Addition-subtraction method 3. 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. Plotting and graphing 6.1 SOLVING A SYSTEM OF TWO LINEAR EQUATIONS page213 Systems of linear equations in the form: This is known as the general form of the equation of a line. page 165 6.1 SOLVING A SYSTEM OF TWO LINEAR EQUATIONS page213 Solving with graphs. Method 1 Graphing (plot) page 213 When the two lines intersect, the system of equations is called independent and consistent. Page 213 When the two lines are parallel, the system of equations is called inconsistent. Page 213 When 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) we shall study is called the addition-subtraction method. (sometimes called the elimination method). Page 214 multiply 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 = -4 X 2y = 5 2 (X 2y = 5) 2 2x 4y = 10 Only 1 equation but both sides. 2x + 3y = -4 -2x 4y = 10 7y = Y = -2 No x. Substitute y = -2 in either original equation. 2 x + 3(-2) = -4 2x + -6 = -4 2x = x = 2 x = 1 multiply each side of one or both equations by some number 2x + 3y = -4 X 2y = 5 2 2x + 3y = x + 6y = -8 4 (X 2y = 5) 4 4x 8y = 20 Both equations. multiply each side of one or both equations by some number 4x + 6y = -8 -(4x 8y) = 20 14y = y = -2 No x. and check. Check by substituting Page 218 Graphing or additions subtraction method. Page 218 Graphing or additions subtraction method. Method 3 Substitution Page 215 The second algebraic method (3 rd method over all) of solving systems of linear equations is called the method of substitution. Page 215 3x + y = 3 2x - 4y = 16 Solve for x or y. (y)3x + y = 3 y = -3x + 3 Page 215 2x - 4y = 162x 4(-3x + 3) = 16 2x + 12x - 12 = 16 14x = 28 x = 2 No y. Page 217 A special case of the substitution method is the comparison method: a = c b = c a = b Page 217 Comparison method: 3x 4 = 5y 6 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 217 Comparison method: 3x 4 = 6 2x 2x +3x 4 = 6 2x + 2x 5x = x = 10 x = 2 No y. Page 218 Except for steps 4, 5, and 6, the steps in approaching the solution of a problem stated verbally are the same as for problems involving only one linear equation. You may wish to review Section 1.13 at this time. Page 218 Section 6.1 The equations in the system are not linear, or first-degree, equations. Some nonlinear systems are too difficult for direct solution by the TI-89. The system in the example, if entered as shown, will take about 1 min of computation time before the calculator (incorrectly) concludes that the system has no solution! 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 Substitute b for x.ax + by = ab a(b) + by = ab - a(b) = - ab by = 0 x = B and y = 0 (b, y) Graphical solutions of three linear equations with three unknowns are not used because three-dimensional graphing is required and is not practical by hand. In class exercise. Check End Week 6 Day 1