box-o-sand/math-replay/algebra/solving-for-a-variable.haml

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    %title Solving for a Variable
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    %h2 Solve for $l$:
    :eqnarray
      -2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\
      -2lmn + 8m - 3n + 9 - 9 & = & -5m + 3n - 7 - 9 \\
      -2lmn + 8m - 3n & = & -5m + 3n - 16 \\
      -2lmn + 8m - 3n + 3n & = & -5m + 3n - 16 + 3n \\
      -2lmn + 8m & = & -5m + 3n - 16 + 3n \\
      -2lmn + 8m - 8m & = & -5m + 3n - 16 + 3n - 8m \\
      -2lmn & = & -5m + 3n - 16 + 3n - 8m \\
      \frac{-2lmn}{-2} & = & \frac{-5m + 3n - 16 + 3n - 8m}{-2} \\
      lmn & = & \frac{-5m + 3n - 16 + 3n - 8m}{-2} \\
      \frac{lmn}{mn} & = & \frac{\frac{-5m + 3n - 16 + 3n - 8m}{-2}}{mn} \\
      l & = & \frac{\frac{-5m + 3n - 16 + 3n - 8m}{-2}}{mn} \\
      l & = & \frac{\frac{-5m - 16 + 6n - 8m}{-2}}{mn} \\
      l & = & \frac{\frac{-16 + 6n - 13m}{-2}}{mn} \\
      l & = & \frac{8 - 3n - \frac{13m}{-2}}{mn} \\

    :eqnarray
      -2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\
      -2lmn + 8m - 3n + 9 - 9 & = & -5m + 3n - 7 - 9 \\
      -2lmn + 8m - 3n & = & -5m + 3n + 2 \\
      -2lmn + 8m - 3n + 3n & = & -5m + 3n + 2 + 3n \\
      -2lmn + 8m & = & -5m + 6n + 2 \\
      -2lmn + 8m - 8m & = & -5m + 6n + 2 - 8m \\
      -2lmn & = & -13m + 6n + 2 \\
      \frac{-2lmn}{-2} & = & \frac{-13m + 6n + 2}{-2} \\
      lmn & = & \frac{-13m}{-2} + \frac{6n}{-2} + \frac{2}{-2} \\
      lmn & = & \frac{-13m}{-2} + \frac{6n}{-2} - 1 \\
      lmn & = & \frac{-13m}{-2} + \frac{3n}{-1} - 1 \\
      lmn & = & \frac{-13m}{-2} - 3n - 1 \\
      \frac{lmn}{mn} & = & \frac{\frac{-13m}{-2} - 3n - 1}{mn} \\
      l & = & \frac{\frac{-13m}{-2} - 3n - 1}{mn} \\

    :eqnarray
      -2lmn + 8m - 3n + 9 & = & -5m + 3n - 7 \\
      -9 + -2lmn + 8m - 3n + 9 & = & -9 + -5m + 3n - 7 \\
      -2lmn + 8m - 3n & = & -9 + -5m + 3n - 7 \\
      -2lmn + 8m - 3n & = & -5m + 3n - 7 - 9 \\
      -2lmn + 8m - 3n & = & -5m + 3n + -7 + -9 \\
      -2lmn + 8m - 3n & = & -5m + 3n + -16 \\
      -2lmn + 8m - 3n + 3n & = & -5m + 3n + -16 + 3n \\
      -2lmn + 8m & = & -5m + -16 + 6n \\
      -2lmn + 8m - 8m & = & -5m + -16 + 6n - 8m \\
      -2lmn & = & -13m + 6n + -16 \\
      \frac{-2lmn}{-2mn} & = & \frac{-13m + 6n + -16}{-2mn} \\
      l & = & \frac{-13m + 6n + -16}{-2mn} \\
      l & = & \frac{13m - 6n + 16}{2mn} \\

    %h2 Solve for $x$:
    :eqnarray
      -2xy + 5xz - 5x + 10 & = & -6y - 2 \\
      -2xy + 5xz - 5x + 10 - 10 & = & -6y - 2 - 10 \\
      -2xy + 5xz - 5x & = & -6y + -2 + -10 \\
      -2xy + 5xz - 5x & = & -6y + -12 \\
      -2xy + 5xz - 5xz - 5x & = & -6y - 12 - 5xz \\
      -2xy - 5x & = & -6y - 12 - 5xz \\
      -2xy + 2xy - 5x & = & -6y - 12 - 5xz + 2xy \\
      -5x & = & -6y - 12 - 5xz + 2xy \\
      -5x \over -5 & = & -6y - 12 - 5xz + 2xy \over -5x \\
      x & = & -6y - 12 - 5xz + 2xy \over -5x \\
      x & = & \frac{-6y}{-5x} - \frac{12}{-5x} - \frac{5xz}{-5x} + \frac{2xy}{-5x} \\

    :eqnarray
      -2xy + 5xz - 5x + 10 & = & -6y - 2 \\
      -2xy + 5xz - 5x + 10 - 10 & = & -6y - 2 - 10 \\
      -2xy + 5xz - 5x & = & -6y + -2 + -10 \\
      -2xy + 5xz - 5x & = & -6y - 12 \\
      -2xy + 5xz - 5x \over 5z & = & -6y - 12 \over 5z \\
      -2xy + x - 5x & = & -6y - 12 \over 5z \\
      -2xy + x + -5x & = & -6y - 12 \over 5z \\
      -2xy + -4x & = & -6y - 12 \over 5z \\
      -2xy - 4x & = & -6y - 12 \over 5z \\
      -2xy - 4x \over -2y & = & -6y - 12 \over -2y + 5z \\
      -x - 4x \over -2y & = & -6y - 12 \over -2y + 5z \\
      -x + -4x & = & -6y - 12 \over -2y + 5z \\
      -5x & = & -6y - 12 \over -2y + 5z \\
      -5x \over -5 & = & -6y - 12 \over -2y + 5z - 5 \\
      x & = & -6y - 12 \over -2y + 5z - 5 \\
      x & = & 6y + 12 \over 2y - 5z + 5 \\

    :eqnarray
      -2xy + 5xz - 5x + 10 & = & -6y - 2 \\
      -2xy + 5xz - 5x + 10 - 10 & = & -6y - 2 - 10 \\
      -2xy + 5xz - 5x & = & -6y - 12 \\
      x \cdot (-2y + 5z - 5) & = & -6y - 12 \\
      x \cdot (-2y + 5z - 5) \over -2y + 5z - 5 & = & -6y - 12 \over -2y + 5z - 5 \\
      x & = & -6y - 12 \over -2y + 5z - 5 \\
      x & = & 6y + 12 \over 2y - 5z + 5 \\

    %h2 Solve for $m$:
    :eqnarray
      -8m - 4n - 8p - 5 & = & -5n - p + 6 \\
      -8m - 4n - 8p - 5 + 5 & = & -5n - p + 6 + 5 \\
      -8m - 4n - 8p & = & -5n - p + 11 \\
      -8m - 4n - 8p + 8p & = & -5n - p + 11 + 8p \\
      -8m - 4n & = & -5n + -p + 11 + 8p \\
      -8m - 4n & = & -5n + 11 + 7p \\
      -8m - 4n + 4n & = & -5n + 11 + 7p + 4n \\
      -8m & = & -n + 11 + 7p \\
      -8m & = & -n + 11 + 7p \\
      -8m \over -8 & = & -n + 11 + 7p \over -8 \\
      m & = & -n + 11 + 7p \over -8 \\
      m & = & n - 11 - 7p \over 8 \\
      m & = & n - 7p - 11 \over 8 \\

    %h2 Solve for $x$ (again):
    :eqnarray
      7xy + 4xz + x + 1 & = & y + 6 \\
      7xy + 4xz + x + 1 - 1 & = & y + 6 - 1 \\
      7xy + 4xz + x & = & y + 5 \\
      x \cdot (7y + 4z + 1) & = & y + 5 \\
      x \cdot (7y + 4z + 1) \over 7y + 4z + 1 & = & y + 5 \over 7y + 4z + 1 \\
      x & = & y + 5 \over 7y + 4z + 1 \\

    %h2 Solve for $v$:
    :eqnarray
      3vw - 2vx + 10v + 1 & = & w + 4 \\
      3vw - 2vx + 10v + 1 - 1 & = & w + 4 - 1 \\
      3vw - 2vx + 10v & = & w + 3 \\
      v \cdot (3w - 2x + 10) & = & w + 3 \\
      v \cdot (3w - 2x + 10) \over 3w - 2x + 10 & = & w + 3 \over 3w - 2x + 10 \\
      v & = & w + 3 \over 3w - 2x + 10 \\