Solving Equations help.?

1. The product of two consecutive positive integers is 11 more than their sum. Find the integers.

2. Solve.

v^2 = 18 - 3v

3. Solve by the substitution method.

15x - 20y = -1

15y = 2 + 5x

3. Solve by the elimination method.

8x - y = 68

x + 3y = 21

4. Solve by the substitution method.

15x - 10y = -5

15y = 4 + 5x

5. Solve the system of equations by the substitution.

0.06x - 0.01y = 0.6

0.22x + 0.13y = 12.2

6. A motel clerk counts his $1 and $10 bills at the end of a day. He finds that he has a total of 59 bills having a combined monetary value of $239. Find the number of bills of each denomination that he has.

the clerk has ____ ones and ___ tens.

1 Answer

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  • 9 years ago
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    1)

    the first integer is = x

    the second integer is ( x + 1 )

    the product as x (x + 1)

    their sum x + (x + 1) -----> 2x + 1

    consecutive positive integers is 11 more than their sum

    x (x + 1) = 11 + 2x + 1

    x^2 + x = 12 + 2x

    x^2 + x - 12 - 2x = 0

    x^2 - x - 12 = 0

    (x - 4)(x + 3) = 0 =====> x = -3 & 4

    ==========

    2)

    v^2 = 18 - 3v

    v^2 + 3v - 18 = 0

    (v + 6)(v - 3) = 0 =====> v = -6 & 3

    =========

    3)

    15x - 20y = -1 ====> 15x = 20y - 1 =====> x = (20/15) y - (1/15) ===> x = (4/3) y - (1/15)

    substitute it into :

    15y = 2 + 5 * ( (4/3) y - (1/15) )

    15y = 2 + (20/3) y - (1/3)

    15y - (20/3) y = (5/3)

    (25/3) y = (5/3)

    25 y = 5

    y = 1/5

    when y = 1/5

    x = (4/3) (1/5) - (1/15)

    x = (4/15) - (1/15)

    x = (3/15)

    x = 1/5

    ============

    3)

    8x - y = 68

    x + 3y = 21 ===> * -8

    8x - y = 68

    +

    -8x - 24y = -168

    -----------------------

    0 - 25y = -100 ======> y = 4

    when y = 4

    8x - 4 = 68

    8x = 68 + 4

    8x = 72

    x = 9

    ===========

    4)

    15x - 10y = -5 ====> 15x = 10y - 5 ====> x = (10/15) y - (5/15) ====> x = (2/3) y - (1/3)

    substitute in:

    15y = 4 + 5 * ( (2/3) y - (1/3) )

    15y = 4 + (10/3) y - (5/3)

    15y - (10/3) y = (7/3)

    (35/3) y = (7/3)

    35 y = 7

    y = 1/5

    when y = 1/5

    x = (2/3) (1/5) - (1/3)

    x = (2/15) - (1/3)

    x = -1/5

    =============

    5)

    (6/100) x - (1/100)y = 6/10

    (22/100)x + (13/100)y = 122/10

    6x - y = 60 =====> * 13

    22x + 13y = 1220

    78x - 13y = 780

    +

    22x + 13y = 1220

    -----------------------

    100x + 0 = 2000 ====> x = 20

    when x = 20

    6*20 - y = 60

    120 - 60 = y

    60 = y

    ============

    6)

    $1 ------- call it -----> x

    $10------ call it -----> y

    has a total of 59 bills -----> x + y = 59

    combined monetary value of $239

    1x + 10y = 239

    x + y = 59

    - <-- minus

    x + 10y = 239

    --------------------

    0 - 9y = -180 =====> y = 20

    when y = 20

    x + 20 = 59

    x = 59 - 20

    x = 39

    ==========

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