A first number plus twice a second number is 10. Twice the first number plus the second totals 35. F

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  1. math_celebrity

    math_celebrity Administrator Staff Member

    A first number plus twice a second number is 10. Twice the first number plus the second totals 35. Find the numbers.

    The phrase a number means an arbitrary variable
    A first number is written as x
    A second number is written as y

    Twice a second number means we multiply y by 2:
    2y

    A first number plus twice a second number:
    x + 2y

    A first number plus twice a second number is 10 means we set x + 2y equal to 10:
    x + 2y = 10

    Twice the first number means we multiply x by 2:
    2x

    Twice the first number plus the second:
    2x + y

    Twice the first number plus the second totals 35 means we set 2x + y equal to 35:
    2x + y = 35

    Therefore, we have a system of two equations:
    1. x + 2y = 10
    2. 2x + y = 35
    Since we have an easy multiple of 2 for the x variable, we can solve this by multiply the first equation by -2:
    1. -2x - 4y = -20
    2. 2x + y = 35
    Because the x variables are opposites, we can add both equations together:
    (-2 + 2)x + (-4 + 1)y = -20 + 35

    The x terms cancel, so we have:
    -3y = 15

    To solve this equation for y, we type it in our search engine and we get:
    y = -5

    Now we substitute this y = -5 into equation 2:
    2x - 5 = 35

    To solve this equation for x, we type it in our search engine and we get:
    x = 20
     

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