-
'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 -3x = -3x 2x - 2 = 8 -
'A student' gets all constants on the right side of the equation.
2x - 2 = 8 + 2 = + 2 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 + 6 = + 6 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 -x = -x 2x - 10 = 6 -
Joe gets the constants together on the right side.
2x - 10 = 6 + 10 = + 10 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 +4 = +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 -1.76x = -1.76x -3.819 = 0.07x - 1.012 -
Joe says, "Let's assemble those constants on the left."
-3.819 = 0.07x - 1.012 +1.012 = + 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