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
  • \(\frac{x}{3} = 25\)
• Multiply both sides by 3
  • \(\require{color} {\color{red}{3}} \cdot \frac{x}{3} = {\color {red}{3}} \cdot 25\)
• Write 3 as \(\frac31\)
  • \(\frac31 \cdot \frac{x}{3} = 3 \cdot 25\)
• Do the multiplications
  • \(\frac{3x}{x} = 75\)
• Simplify \(\frac{3x}{3}\) by removing the common factor of 3 in the numerator and denominator: \(\require{cancel} \frac{\cancel{3}^1x}{\cancel{3}_1} = x\)
  • \(1x = 75\)
• The coefficient 1 need not be written since \(1x = x\)
  • \(x = 75\)

If we substitute 75 for x in \(\frac{x}{3} = 25\), we obtain the true statement 25 = 25.  This verifies that 75 is the solution.