Solve the following equation

Explanation

We will use a property of equality to isolate the variable on one side of the equation.  To solve the original equation, we want to find a simpler equivalent equation of the form x = a number, whose solution is obvious.

Solution

• This is the equation to solve
  • \(\frac23x = 6\)
• To undo the multiplication by \(\frac23\), multiply both sides by the reciprocal of \(\frac23\).
  • \(\require{color} {\color{red}{\frac32} \cdot} \frac23x = {\color{red}{\frac32} \cdot}  6\)
• Use the associative property of multiplication to group \(\frac32\) and \(23\)
  • \(\left(\frac32 \cdot \frac23 \right)x = \frac32 \cdot 6\)
• On the left side, the product of a number and its reciprocal is 1: \(\frac32 \cdot \frac23 = 1\).  On the right side, \(\frac32 \cdot 6 = \frac{18}{2} = 9\).
  • \(1x = 9\)
• The coefficient 1 need not be written since \(1x = x\)
  • \(x = 9\)

To check the answer substitute 9 for x in the original equation

• \(\frac23(9) = 6\)
• 6 = 6

Since the statement 6 = 6 is true, 9 is the solution of \(\frac23x = 6\)