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Systems of equations lesson 5

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  • 1. SYSTEMS OF EQUATIONS
    MATH10
    ALGEBRA
    Systems of Equations(Algebra and Trigonometry, Young 2nd Edition, page 874-904)
  • 2. Week 6 Day 1
    GENERAL OBJECTIVE
    At the end of the chapter the students are expected to:
    • Solve systems of equations in two variables with the substitution method and the elimination method.
    • 3. Graph systems of linear equations.
    • 4. Understand that systems of linear equations may have one solution, no solution, or infinitely many solutions.
    • 5. Solve systems of equations in three variables employing combination of the elimination and substitution methods.
  • SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES
  • 6. Week 6 Day 1
    TODAY’S OBJECTIVE
    At the end of the lesson the students are expected to:
    • Tosolve systems of linear equations in two variables using the substitution method.
    • 7. To solve systems of linear equations in two variables using the elimination method.
    • 8. To solve systems of linear equations in two variables by graphing.
    • 9. Understand that a system of linear equations has either one solution, no solution or infinitely many solution.
  • Week 6 Day 1
    DEFINITION
    SYSTEMS OF EQUATIONS
    A system of equationsis a set of equations that involve the same variables. To solve a system ofequations means to find the solution that satisfies both equations.
    Example:
    Solutions are given as an ordered pair of the form (x,y) .
  • 10. Week 6 Day 1
    Example:
    y
    6
    3x-y=11
    x+2y=6
    Check
    x
    -4
    INTERPRETATION
  • 11. Week 6 Day 1
    THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
    y
    6
    1. Independent
    System
    One solution
    x
    -4
    Lines have different slopes.
  • 12. Week 6 Day 1
    THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
    y
    6
    2. Inconsistent
    System
    No solution
    x
    -4
    Lines are parallel (same slopes and different y - intercepts.)
  • 13. Week 6 Day 1
    THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
    y
    6
    3. Dependent
    System
    Infinitely many
    solution
    x
    -4
    Lines coincide (same slopes and same y – intercepts).
  • 14. Week 6 Day 1
    METHODS OF SOLVING SYSTEMS OF LINEAR EQUATIONS
    Three methods of solving systems of linear equations:
    • Algebraic Methods – used to find exact solutions.
    substitution method
    elimination method
    • Graphing Method – typically used to give a visual interpretation and confirmation of the solution.
  • Week 6 Day 1
    SUBSTITUTION METHOD
    STEPS:
    Solve one of the equations for one variable in terms of the other variable.
    Substitute the expression found in step 1 into the other equation. The result is an equation in one variable.
    Solve the equation obtained in step 2.
    Back substitute the value found in step 3 onto the expression found in step 1.
    Check that the solution satisfies both equations.
  • 15. Week 6 Day 1
    DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS ONE SOLUTION
  • 16. Week 6 Day 1
    DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS NO SOLUTION
  • 17. Week 6 Day 1
    DETERMINING BY SUBSTITUTION THAT A SYSTEM HAS INFINITE SOLUTION
  • 18. Week 6 Day 1
    ELIMINATION METHOD
    STEPS:
    Multiply the coefficients of one or both of the equations so that one of the variables will be eliminated when two equations are added.
    Eliminate one of the variables by adding the expression found in Step 1 to the other equation. The result is an equation in one variable.
    Solve the equation obtained in Step 2.
    Back substitute the value found in Step 3 into either of the original equation.
    Check that the solution satisfies both equations.
  • 19. Week 6 Day 1
    DETERMINING BY ELIMINATION THAT A SYSTEM HAS ONE SOLUTION
  • 20. Week 6 Day 1
    DETERMINING BY ELIMINATION THAT A SYSTEM HAS NO SOLUTION
  • 21. Week 6 Day 1
    DETERMINING BY ELIMINATION THAT A SYSTEM HAS INFINITELY MANY SOLUTION
  • 22. Week 6 Day 1
    GRAPHING METHOD
    STEPS:
    Write the equations in the slope-intercept form.
    Graph the lines.
    Identify the points of intersection.
    Check that the solution satisfies both equations.
    OR
    Solve for the x and y intercepts.
    Graph the lines.
    Identify the points of intersection.
    Check that the solution satisfies both equations.
  • 23. Week 6 Day 1
    DETERMINING BY GRAPH THAT A SYSTEM HAS ONE SOLUTION, NO SOLUTION OR INFINITELY MANY SOLUTION
  • 24. Week 6 Day 1
    IDENTIFYING WHICH METHOD TO USE
    Given any system of linear equations in two variables, any of the three
    methods ( substitution, elimination, or graphing) can be utilized.
    • Elimination is preferred if it is easy to eliminate a variable by adding
    multiples of two equations.
    • Use substitution if there is no obvious elimination.
    • 25. Use graphing to confirm the solution(s) found using either elimination
    or substitution.
    EXAMPLE
    State which of the two algebraic methods (elimination or substitution)
    would be the preferred method to solve each system of linear equations.
  • 26. Week 6 Day 1
    SUMMARY
    Three types of solutions to systems of linear equations:
    • One solution – the system is called an independent system
    - the lines formed are intersecting lines
    • No solution - the system is called inconsistent system
    - the lines formed are parallel lines
    • Infinitely many solutions –the system is called dependent system
    - the lines formed coincide.
    Three methods of solving systems of linear equations:
    • Algebraic Methods – used to find exact solutions.
    substitution method
    elimination method
    • Graphing Method – typically used to give a visual interpretation and confirmation of the solution.
  • Week 6 Day 2
    SYSTEMS OF LINEAR EQUATIONS IN THREE VARIABLES
  • 27. Week 6 Day 2
    TODAY’S OBJECTIVE
    At the end of the lesson the students are expected to:
    • To understand that a graph of linear equation in three variables correspond to a plane.
    • 28. To identify three types of solutions: one solution, no solution or infinitely many solutions.
    • 29. To solve systems of linear equations in three variables using the combination of both elimination method and the substitution method.
  • Week 6 Day 2
    DEFINITION
    SYSTEMS OF EQUATIONS IN THREE VARIABLES
    • A linearequation in three variables x, y, and z is given by
    Ax +By +C = D
    where A, B, C, and D re real numbers that are not all equal to zero.
    • All three variables have degree equal to one, which is why this is called equation in three variables .
    • 30. The graph of any equation in three variables requires three dimensional coordinate system.
    • 31. In two variables, the graph of a linear equation is a line, while in three variables the graph of a linear equation is a plane which can be thought of as an infinite sheet of paper.
    • 32. Solutions are given as an ordered pair of the form (x,y,z)
  • Week 6 Day 2
    THREE TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS IN THREE VARIABLES
    Independent - one solution
    Dependent - infinitely many solutions
    Inconsistent - no solution
  • 33. Week 6 Day 2
    One Solution
    Solution
  • 34. Week 6 Day 2
    No Solution
    or
  • 35. Week 6 Day 2
    Infinitely Many Solutions
    Solution
    (line of intersection)
  • 36. Week 6 Day 2
    SOLVING SYSTEMS OF LINEAR EQUATIONS IN THREE VARIABLES USING ELIMINATION AND SUBSTITUTION
    STEPS
    Reduce the system of three equations in three variables to two
    equations in two (of the same) variables by applying elimination.
    Solve the resulting system of two linear equations in two variables
    by applying elimination or substitution.
    Substitute the solution in Step 2 into any of the original equations
    and solve for the third variable.
    4. Check that the solution satisfies all three original equations.
  • 37. Week 6 Day 2
    EXAMPLE
  • 38. Week 6 Day 2
    SOLVING SYSTEMS OF TWO LINEAR EQUATIONS IN THREE VARIABLES
    • Two linear equations in three variables will always correspond to
    two planes in three dimensions.
    • There are two possibilities
    No solution if the two planes are parallel
    Infinitely many solutions if the two planes intersect in a line.
  • 39. Week 6 Day 2
    No solution if the two planes are parallel
    z
    y
    x
  • 40. Week 6 Day 2
    Infinitely many solutions if the two planes intersect in a line.
    z
    solution
    y
    x
  • 41. Week 6 Day 2
    EXAMPLE
  • 42. Week 6 Day 2
    SUMMARY
    • Graphs of linear equations in two variables are lines, whereas graphs of linear equations in three variables are planes.
    • 43. Systems of linear equations in three variables have one of the three outcomes:
    One solution (point)
    No solution (no intersection of all three planes)
    Infinitely many solutions (planes intersect along a line)
    • When the solution to a system of three linear equations is a line in three dimensions, we use parametric representations to express the solution.
  • Week 6 Day 3
    TODAY’S OBJECTIVE
    At the end of the lesson the students are expected to:
    • To solve systems of nonlinear equations.
    • 44. To solve application problems involving systems of equations in two and in three variables.
  • Week 6 Day 3
    QUADRATIC SYSTEMS IN TWO VARIABLES
    The most general form of a quadratic equation in the variables x and y is
    The graphs of these equations are circles and the conic sections which
    are to be discussed in analytic geometry.
  • 45. Week 6 Day 3
    QUADRATIC SYSTEMS IN TWO VARIABLES
    2. Two Quadratics
    1. One Linear, One Quadratic
    3.Two Quadratics, all terms containing the variable are of second degree
    College Algebra Revised edition, Catalina D. Mijares page 241-250
  • 46. Week 6 Day 3
    QUADRATIC SYSTEMS IN TWO VARIABLES
    4. Symmetric Quadratic Equation
    5.Other types which does not fall on the previous types.
    College Algebra Revised edition, Catalina D. Mijares page 241-250
  • 47. Week 6 Day 3
    APPLICATION INVOLVING SYSTEMS OF LINEAR EQUATIONS
    Week 6 Day 3 Application Involving Systems of Linear Equations (Algebra and Trigonometry, Young 2nd Edition, page 886-891 and 899-904).
  • 48. Week 6 Day 3
    RECALL
    Start
    A
    Read and analyze the problem
    Make a diagram or sketch if possible
    Solve the equation
    Determine the unknown quantity.
    Check the solution
    Set up an equation,
    assign variables to
    represent what you
    are asked to find.
    no
    Is the unknown
    solved?
    no
    yes
    yes
    Did you set up
    the equation?
    A
    End
  • 49. Week 6 Day 3
    APPLICATION
    1. Upon graduation with a degree of management information systems(MIS), you decide to work for a company that buys data from the states’ department of motor vehicles and sells to banks and car dealerships customized reports detailing how many cars at each dealership are financed through particular banks. Autocount Corporation offers you a $15,000 base salary and a 10% commission on your total annual sales. Polk Corporation offers you a base salary of $30,000 plus a 5% commission on your total annual sales. How many total sales would you have to make per year to make more money at Autocount? (# 59 page 890)
  • 50. Week 6 Day 3
    APPLICATION
    2. A mechanic has 340 gallons of gasoline and 10 gallons of oil to make gas/oil mixtures. He wants one mixture to be 4% oil and the other mixture to be 2.5% oil. If he wants to use all of the gas and oil, how many gallons of gas and oil are in each of the resulting mixtures? (# 58 page 890)
    3. A direct flight on Delta Airlines from Atlanta to Paris is 4000 miles and takes approximately 8 hours going East (Atlanta to Paris) and 10 hours going West ( Paris to Atlanta). Although the plane averages the same airspeed, there is a headwind while traveling west and a tailwind while travelling east resulting in different airspeeds. What is the average airspeed of the plane and what is the average wind speed ? (# 63 page 890)
  • 51. Week 6 Day 3
    APPLICATION
    Suppose you’re going to eat only Subway sandwiches for a week (7 days) for lunch and dinner (total o0f 14 meals).
    4. Your goal is a total of 4840 calories and 190 grams of fat. How many of each sandwich would you eat that week to obtain this goal? ( #33 page 901)
  • 52. Week 6 Day 3
    APPLICATION
    5. Tara and Lamar decide to place $20,000 of their savings into investments. They put some in a money market account earning 3% interest, some in a mutual fund that has been averaging 7% a year, and some in a stock that rose 10% last year. If they put $6,000 more in the money market than in the mutual fund and the mutual fund and stocks have the same growth in the next year as they did in the previous year , they will earn $1,180 in a year. How much money did they put in each of the three investments? (# 39 page 902)