Math Analysis Notes Mrs. Atkinson 1

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1 Name: Math Analysis Chapter 7 Notes Day 6: Section 7-1 Solving Systems of Equations with Two Variables; Sections 7-1: Solving Systems of Equations with Two Variables Solving Systems of equations with two variables by Substitution: 1. Solve either of the equations for one variable in terms of the other, if necessary. (If possible, solve a variable whose coefficient is 1 or 1 to avoid working with fractions).. Substitute the expression found in step 1 into the other equation. This will result in an equation with one variable. 3. Solve the equation containing one variable. 4. Back-substitute the value found in step 3 into one of the original equations. Simplify and find the value of the remaining variable. 5. Check the proposed solution in both of the system s given equations. Practice: 1. Solve by substitution method: 3x + y = 4 x + y = 1 Solving Systems of equations with two variables by Elimination: 1. If necessary, rewrite both equation in the form Ax + By = C.. If necessary, multiply either equation or both equations by appropriate nonzero numbers so that the sum of the x-coefficients or the sum of the y-coefficients is Add the equations in step. The sum is an equation in one variable. 4. Solve the equation in one variable. 5. Back-substitute the value obtained in step 4 into either of the given equation and solve for the other variable. 6. Check the proposed solution in both of the system s given equations. Practice : Solve by the elimination method (aka the addition method): 4x + 5y = 3 x 3y = 7 Math Analysis Notes Mrs. Atkinson 1

2 Remember: If solving the system by either method you get a true statement without a variable in the equation the system has infinitely many solutions which we will now be defining that solution. Practice 3: Solve the system using any method: x 4y = 8 x= 7y 17 5x 0y = 40 5y = 17 3x Applications Strategy for Problem Solving Using Systems of Equations Step 1: Read the problem carefully. Attempt to state the problem in your own words and state what the problem is looking for. Use variables to represent unknown quantities. Step : Write a system of equations that models the problem s conditions. Step 3: Solve the system and answer the problem s question. Practice 4: A chemist needs to mix 18% acid solution with 45% acid solution to obtain 1 liters of 36% acid solution. How many liters of each of the acid solutions must be used? Math Analysis Notes Mrs. Atkinson

3 Revenue and Cost Functions A company produces and sells x units of a product. Revenue Function: R(x) = (price per unit sold)x Cost Function: C(x) = fixed cost + (cost per unit produced)x The break-even point is the intersection of the graphs of the revenue and cost functions (aka solution of the system). The x-coordinate of the point reveals the number of units that a company must produce and sell so that money coming in, the revenue, is equal to the money going out, the cost. The y-coordinate of the break-even point gives the amount of money coming in and going out. Practice 5: A company that manufactures running shoes has a fixed cost of $300,000. Additionally, it costs $30 to produce each pair of shoes. They are sold at $80 per pair. a) Write the cost function, C, of producing x pairs of running shoes. b) Write the revenue function, R, from the sale of x pairs of running shoes. c) Determine the break-even point. Describe what this means. Math Analysis Notes Mrs. Atkinson 3

4 Day 7: Section 7- Solving Systems of Equations with Three Variables. Section 7-: Systems of Linear Equations in Three Variables. Solving Linear Systems in Three Variables by Eliminating Variables. 1. Reduce the system to two equations in two variables. This is usually accomplished by taking two different pairs of equations and using the elimination method to eliminate the same variable from both pairs.. Solve the resulting system of two equations in two variables using elimination or substitution. The result is an equation in one variable that gives the value of that variable. 3. Back-substitute the value of the variable found in step into either of the equations in two variables to find the value of the second variable. 4. Use the values of the two variables from steps and 3 to find the value of the third variable by back-substitution into one of the original equations. Practice 6: Solve the system: x + 4y z = 0 3x + y + z = 8 x 3y + z = 16 Math Analysis Notes Mrs. Atkinson 4

5 Quadratic regression analysis: Application: Writing the equation of a quadratic equation knowing three points that the equation passes through. General form of a quadratic equation is: y = ax + bx + c In order to find the quadratic equation passing through 3 given points write the system of equations that models to given information and solve for the coefficients a, b, and c. Practice 7: Find the quadratic function whose graph passes through the points (1, 4), (, 1) and (3, 4). Math Analysis Notes Mrs. Atkinson 5

6 Day 8: 7-3 Partial fraction decomposition After completing today s notes you should be able to do the following: P 1. Decompose, where Q has only distinct linear factors Q P. Decompose, Q P 3. Decompose, Q P 4. Decompose, Q where Q has repeated linear factors where Q has a non-repeated prime quadratic factor where Q has a non-repeated prime quadratic factor The reason we want to be able to find the partial fraction of decomposition of a rational expression is to be able to study rates of change. Calculus is the study of rates of change and partial fraction of decomposition is an algebraic technique used to find a function if its rate of change is known. Steps in Partial Fraction Decomposition 1. Set up the partial fraction decomposition with the unknown constants A, B, C, etc., in the numerator of the decomposition.. Multiply both sides of the resulting equation by the least common denominator. 3. Simplify the right-side of the equation. 4. Write both sides in descending powers, equate coefficients of like powers of x, and equate constant terms. 5. Solve the resulting linear system for A, B, C, etc. 6. Substitute the values for A, B, C, etc., into the equation in step 1 and write the partial fraction decomposition. Px ( ) The Partial Fraction Decomposition of : Q(x) Has Distinct Linear Factors. Qx ( ) The form of the partial fraction decomposition for a rational expression with distinct linear factors in the denominator is: Px ( ) A B = (until number of linear factors are matched) ( ax+ b)( ax+ b)( ax+ b)...( ax+ b) ax+ b ax+ b n n 1 1 Practice 1: Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (a) 11x 10 ( x )( x+ 1) (b) 1 x(x+ 3) Math Analysis Notes Mrs. Atkinson 6

7 Practice : Write the partial fraction decomposition of: 3x + 50 ( x 9)( x+ ) Px ( ) The Partial Fraction Decomposition of : Q(x) Has Repeated Linear Factors. Qx ( ) The form of the partial fraction decomposition for a rational expression containing the linear factor (ax + b) occurring n times as its denominator is: Px ( ) A B? = ( ) n (keep adding constant terms until you match the power of the linear factor) n ax + b ax + b ( ax + b) ( ax + b) Practice 3: Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x + 3x+ 5 x (a) (b) 3 ( x+ )( x 3) ( x 1) ( x+ 1) Math Analysis Notes Mrs. Atkinson 7

8 Practice 4: Write the partial fraction decomposition of: x + x+ 7 xx ( 1) Px ( ) The Partial Fraction Decomposition of : Q(x) Has Non-Repeated, Prime Quadratic Factor. Qx ( ) If ax + bx + c is prime quadratic factor of Q(x), the partial fraction decomposition will contain a term of the form Ax+ B ax + bx + c Practice 5: Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. 3 x x+ 8 5x + x + x+ 4 (a) (b) ( x+ 1)( x + x+ ) ( x + 1)( x + 3x+ 5) Practice 6: Write the partial fraction decomposition of: 5x + 6x+ 3 ( x ) ( x 1) + 1 Math Analysis Notes Mrs. Atkinson 8

9 Px ( ) The Partial Fraction Decomposition of : Q(x) Has a Prime, Repeated Quadratic Factor. Qx ( ) The form of the partial fraction decomposition for a rational expression containing the prime factor ax + bx + c occurring n times as its denominator is: Px ( ) Ax+ B Dx+ C? = ( ) n (keep writing linear equations as numerator) n ax + bx + c ax + bx + c ( ax + bx + c) ( ax + bx + c) Practice 7: Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (a) 3 x + x+ 3 ( x+ 1)( x + 1) 3 (b) 3 xx ( + x+ 4) Practice 8: Write the partial fraction decomposition of: 3 x x x ( x x+ 3) Math Analysis Notes Mrs. Atkinson 9

10 Practice 9: Write the partial fraction decomposition of: 4x + 3x x 8 Math Analysis Notes Mrs. Atkinson 10

11 Day 9: Section 7-4 Systems of Nonlinear Equations in Variables; After completing today s notes you should be able to do the following: 1. Solve nonlinear systems by substitution or elimination methods. Graph a system of inequalities. 3. Solve applied problems involving systems of inequalities Remember that the same steps for solving a nonlinear system is the same as if it was linear. See page 5 for steps to solve systems of equations: Section 7-4 Systems of Nonlinear Equations x = y 1 Practice 1: Solve by substitution method: 4x y = 1 Practice : Solve by substitution method: x+ y = 0 ( x 1) + ( y 1) = 5 Math Analysis Notes Mrs. Atkinson 11

12 Practice 3: Solve: 3x + y = 35 4x + 3y = 48 Practice 4: Find the length and width of a rectangle whose perimeter is 0 feet and whose area is 1 square feet. Practice 5: The difference between the square of two numbers is 3. Twice the square of the first number increased by the square of the second number is 9. What are the numbers? Math Analysis Notes Mrs. Atkinson 1

13 Day 30: Section 7-5 Systems of Inequalities Section 7-5 Systems of Inequalities x+ y< Practice 6: Graph the solution set of the system: x < 1 y > 3 Remember graphing equations of circles: (x h) + (y k) = r where (h, k) = center and r = radius. We are now going to graph inequalities that are circles. ( x 1) + ( y+ 1) < 5 Practice 7: Graph the solution set of the system: ( x 1) + ( y+ 1) 16 Math Analysis Notes Mrs. Atkinson 13

14 x + y 1 Practice 8: Graph the solution set of the system: y x > 0 Math Analysis Chapter 7 Review Worksheet Please do the following problems on a separate sheet of paper. 1. A small business has an initial investment of $5000. The unit cost of product is $1.60, and the selling prices $ (a) Write cost and revenue functions for x units produced and sold. (b) Find the break even point and describe what this means.. A chemist needs to mix a solution that is 34% silver nitrate with one that is 4% silver nitrate to obtain 100 milliliters of mister that is 7% silver nitrate. How many milliliters of each of the solutions must be used? 3. Find the quadratic equation in standard from whose graph passes through the points (1, 4), (3, 0), and (, 5). In 4-7, write the partial fraction decomposition of each rational expression. 4. x ( x 3)( x+ ) x 6 5. ( x 1)( x ) 6. 7x 7x+ 3 ( x 3)( x + 4) x + 5x + 7x+ 1 ( x + x+ 1) Math Analysis Notes Mrs. Atkinson 14

15 In 8-10, solve each system by the method of your choice. 8. 5y = x 1 9. xy 4 = x + 3y = 1 x y = 1 y x = 0 3x 4y = 3 In 11-13, graph the solution set of each system of inequalities or indicate that the system has no solution. 11. x + y y > x 13. y 0 x + y < x + y < 6 x < 5 y < x + 6 x > y < 6 Math Analysis Notes Mrs. Atkinson 15