Homework of an 'A' student who is solving for x

  • 'A' student looks at his first homework problem.

    5x - 2 = 3x + 8
  • 'A' student works to get x on the left side of the equation.

    5x - 2 = 3x + 8 2x - 2 = 8
  • 'A student' gets all constants on the right side of the equation.

    2x - 2 = 8 2x = 10
  • 'A student' divides to get a single unit of x on the left side.

    2x = 10
    2x / 2 = 10 / 2
    x = 5
  • 'A student' decides to start going by the name of "Joe."

  • Joe looks at his second homework problem.

    4x - 2(x + 3) = -18
  • Joe correctly contends with the parentheses first.

    4x - 2(x + 3) = -18
    4x - (2x + 6) = -18
    4x - 2x - 6 = -18
  • Joe simplifies like terms on the left side.

    4x - 2x -6 = -18
    2x - 6 = -18
  • Joe isolates the constant on the right side of the equation.

    2x - 6 = -18 2x = -12
  • Joe divides on both sides to get a single unit of x on the left.

    2x = -12
    2x / 2 = -12 / 2
    x = -6
  • Joe is annoyed to discover that problem #3 is full of fractions.

    x/2 - 5/3 = x/6 + 1
  • Joe would really like to get rid of those fractions.

    x/2 - 5/3 = x/6 + 1
    Joe notes that the numbers in the denominators are 2, 3, and 6. Two and three are both multiples of six, and the Lowest Common Denominator is 6.
  • Joe multiplies each side by 6 to eliminate the fractions.

    x/2 - 5/3 = x/6 + 1
    6(x/2) - 6(5/3) = 6(x/6) + 6(1)
    3x - 10 = x + 6
  • Joe subtracts x from each side to get the variable on the left.

    3x - 10 = x + 6 2x - 10 = 6
  • Joe gets the constants together on the right side.

    2x - 10 = 6 2x = 16
  • Joe multiplies each side by two to isolate a single unit of x.

    2x = 16
    2x/2 = 16/2
    x = 8
  • Joe again sees fractions -- with x in the denominator this time!

    12/x = 4(1/2 - 1/x)
  • Joe distributes the 4 on the right side.

    12/x = 4(1/2 - 1/x)
    12/x = 4(1/2) - 4(1/x)
    12/x = 2 - 4/x
  • Joe decides he's tired of having variables in the denominator.

    12/x = 2 - 4/x
    x(12/x) = x(2 - 4/x)
    12 = 2x - 4
  • Joe gets the constants all on the left.

    12 = 2x - 4 16 = 2x
  • Joe divides on both sides to solve in time for Thanksgiving dinner!

    16 = 2x
    16/2 = 2x/2
    8 = x
  • That last problem was hard. Joe checks his answer.

    8 = x
    12/x = 4(1/2 - 1/x)
    12/8 = 4(1/2 - 1/8)
    12/8 = 2 - 1/2
    3/2 = 3/2
  • Joe looks at his last problem and says, "Decimals! Ugh!"

    1.76x - 3.819 = 0.68x (0.5x - 0.44)
  • "Okay, okay," Joe says. "It's time to distribute."

    1.76x - 3.819 = 0.68x + 2.3 (0.5x - 0.44)
    1.76x - 3.819 = 0.68x + 1.15x - 1.012
  • Joe says, "It's time to combine like terms."

    1.76x - 3.819 = 0.68x + 1.15x - 1.012
    1.76x - 3.819 = 1.83x - 1.012
  • "Let's get the x's together over on the right," Joe says.

    1.76x - 3.819 = 1.83x - 1.012 -3.819 = 0.07x - 1.012
  • Joe says, "Let's assemble those constants on the left."

    -3.819 = 0.07x - 1.012 -2,807 = 0.07x
  • Joe says, "Dividing each side by 0.07 will leave a unit of x."

    -2.807 = 0.07x
    -2.807/0.07 = 0.07x/0.07
    -4.01 = x
  • Joe says, "I'm double-checking that."

    x = -40.1
    1.76x - 3.819 = 0.68x + 2.3 (0.5x - 0.44)
    1.76 (-40.1) - 3.819 = 0.68 (-40.1) + 2.3 [(0.5)(-40.1) - 0.44]
    -70.576 - 3.819 = -27.268 + 2.3 (-20.05 - 0.44)
    -74.395 = -27.268 - 46.115 - 1.012
  • Joe: "Whew! It looks like I'm right."

    -74.395 = -27.268 - 46.115 - 1.012
    -74.395 = -74.395