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Systems of Linear Equations - Ahlborn.us

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Systems of Linear Equations - Ahlborn.us

Algebra I Turner

    Chapter 7

Lesson Goal: Solve a system of two linear equations by graphing.

    Definition System of Equations: A system of equations consists of two or more

    equations in two or more variables. Definition Solution of a System of Equations: A solution of a system of equations in

    two variables is an ordered pair that makes every equation in the system true.

    5xy83x2y21. Consider the system: and .

    (6,3)(1,3)a. Is a solution? b. Is a solution?

    (2,2)c. Is a solution?

    x;y7xy32. Consider this system of equations: and . Can you find some

    solutions by trial and error?

     y

    8

    7 6

    5

    4

    x;y7xy33. Solve graphically: and . 3

    2

    1

    x 0 -8 -3 -2 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -1 -1

    -2

    -3

    -4

    -5

    -6

    -7

    -8

    Algebra I Turner Classwork 7

    4. Could a system of linear equations ever have more than one solution? Could it ever

    have no solutions?

    y

    8 2x;3y64x;3y12 and . 5. Solve graphically: 7

    6

    5 4

    3

    2 1

    x 0 -8 -3 -2 1 2 3 4 5 6 7 8 -7 -6 -5 -4 -1 -1

    -2 -3

    -4

    -5 -6

    -7

    -8

    y 18 3y;x96. Solve graphically: and . yx;17 36

    5

    4 3

    2

    1 x 0 4 6 -8 -3 -2 1 2 3 5 7 8 -7 -6 -5 -4 -1 -1

     -2

    -3

    -4

    -5

    -6

    -7

    -8

2

    Algebra I Turner Classwork 7

    y 28 and . 7. Solve graphically: x4yx;137

    6

    5

    4

    3

    2

    1 x 0 1 2 3 4 5 6 7 8 -8 -7 -6 -5 -4 -3 -2 -1 -1

    -2 -3

    -4

    -5 -6

    -7

     -8

    How many solutions would each system have?

    yx;4yx;98. and

    y2x4y3x;69. and

    y6x;2y6x210. and

Lesson Goal: Solve a system of two linear equations by substitution.

    Process Solving a System of Equations by Substitution

     1. Solve one equation for one variable.

     2. Substitute this expression into the other equation.

     3. Solve the resulting equation.

     4. Substitute this value into the expression obtained from step #1.

     5. Write the solution set.

    3

    Algebra I Turner Classwork 7

    2xy97x;3y1 and . 11. Solve by substitution:

    4x;y15xy1012. Solve by substitution: and .

    2x4y72y;x113. Solve by substitution: and .

    4

Algebra I Turner Classwork 7

    5x;3y15x;3y7 and . 14. Solve by substitution:

    Lesson Goal: Solve a word problem using a system of linear equations. 15. A quilt maker sews both large and small quilts. A large quilt requires 8 yards of

    fabric while the small quilt requires 3 yards. How many of each size quilt did she

    make if she used a total of 90 yards of fabric to make 15 quilts?

    16. A store is selling all shoes for $12 a pair and all slippers for $8 a pair. If Linda

    buys a total of 7 items and spends a total of $76, how many pairs of shoes and

    slippers did she buy?

    5

Algebra I Turner Classwork 7

    17. A toy maker produces wooden trains and wooden airplanes. Each train requires 3

    ounces of paint and each airplane requires 5 ounces of paint. The toy maker has a

    gallon can of paint (64 ounces). If he wants to paint 14 toys, how many of each

    can he paint?

    18. A girl has 15 coins, all nickels and dimes, with a total value of $1.20. Find the

    number of each kind of coin.

    Lesson Goal: Solve a system of two linear equations by addition. Review:

    aba;cb;cTheorem Addition Property for Equality: If , then .

    Restatement:

    Theorem Addition Property for Equality: If abcd and , then

    a;cb;d.

    Now consider it written in this form:

    ab

    (;)cd

    a;cb;d

    This is the form of the property that we are going to use in our solutions today.

    6

Algebra I Turner Classwork 7

    xy1x;y7 and . Then check. 19. Solve by addition:

20. Solve by addition: and . 2a;b152a;3b5

    5x3y325x;7y821. Solve by addition: and .

    22. A plumber charges a flat fee to work on a job plus an hourly rate. If he charged

    $150 for a 3 hour job and $300 for an 8 hour job, what was the flat fee and the

    hourly rate?

    7

Algebra I Turner Classwork 7

    Lesson Goal: Solve a system of two linear equations by addition and multiplication.

    23. Solve by linear combinations: and . 3m;4n25mn12

24. Solve by linear combinations: and . 4r14;5t3r;6t9

    7x142y28;4y14x25. Solve by linear combinations: and .

    26. The winning average on a Junior High baseball team was 0.381. If the team had

    won 3 more games, then its winning average would have been 0.524 . Find the

    number of games the team won and the number of games it played.

    8

Algebra I Turner Classwork 7

    27. The difference between three times one number and a smaller number is 23. The

    sum of the smaller and twice the larger number is 27. Name the numbers.

    Lesson Goal: Solve interest problems using a system of linear equations. Using the Language of Investments

    Review:

    InterestPrincipalRateTimeFormula ;Interest:

    Investment problems use a variety of words to describe interest, principal, and rate. Here are some of the words you should associate with each:

     Principal always refers to the starting amount of money that is invested, deposited,

    or borrowed.

     Rate is sometimes referred to as interest, but it is easily identified as the rate

    because it is always expressed as a percentage.

     Interest is the amount of money that is earned by the investment. It can be

    referred to as the yield, dividend, growth, increase, return, earnings, or income. 28. Mr. Kirchner invested $8000, part earning 8% interest and the rest earning 10%

    interest per year. How much did he invest at each rate if his annual yield was $760?

    9

Algebra I Turner Classwork 7

    29. Mrs. Mooney wants to earn at least $288 in annual dividends. She plans to invest a

    certain sum of money at 12% and 3 times as much at 8%. What is the least

    amount she can invest at 12% to accomplish this goal?

    30. Mrs. Miller deposited $1400, part at 9% and the rest at 12% annual interest. If

    both deposits earn the same annual income, how much has been deposited at each

    rate?

    31. Miss Dilmer took out two loans to pay for a car. One loan required her to pay 10%

    annual interest while the other required her to pay 15% annual interest. Miss

    Dilmer borrowed $1400 more at the lower rate than she did at the higher rate. If

    the annual interest on the loan at 10% was $25 less than the interest on the loan at

    15%, how much did she borrow at each rate?

    10

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