• # '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

• # 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

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

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