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 t = a number or a number = t, whose solution is obvious.

Solution

To isolate t on the left side, we use the division property of equality.  We can undo the multiplication by 2 by dividing both sides of the equation by 2.

• This is the equation to solve
  • 2t = 80
• Divide both sides by 2
  • \(\frac{2t}{2} = \frac{80}{2}\)
• On the left side, simplify \(\frac{2t}{2}\) by removing the common factor of 2 in the numerator and denominator:  \(\require{cancel}\frac{\cancel{2}^1t}{\cancel{2}_1} = t\).  On the right side, do the division.
  • 1t = 40
• The coefficient 1 need not be written since 1t = t
  • t = 40

If we substitute 40 for t in 2t = 80, we obtain the true statement 80 = 80.  This verifies that 40 is the solution.  Since division by 2 is the same as multiplication by \(\frac12\), we can also solve 2t = 80 using the multiplication property of equality.  We could also isolate t by multiplying both sides by the reciprocal of 2, which is \(\frac12\):

\(\frac12 \cdot 2t = \frac12 \cdot 80\)