Math CentralQuandaries & Queries


Question from Desiree, a student:

3x -5^2=8


The goal of solving this equation is to manipulate it to produce a sequence of equivalent equations ending with an equation of the form "x = some number". Let me illustrate with the following problem.

Solve $4x + 28 = 2^3.$

The allowable operations which will produce an equivalent equation are

  1. Add the same amount to each side of the equation.

  2. Multiply both sides of the equation by the same amount.

In the equation

\[4x + 28 = 2^3\]

I first want to eliminate the $28$ from the left side. To do so I add $-28$ to both sides of the equation. This yields

\[4x + 28 - 28 = 2^3 - 28\]

Since $2^3 = 8$ this simplifies to

\[4x = -20\]

All that remains is to remove the 4 coefficient and to do this I will multiply both sides by $\large \frac14 .$ The result is

\[\frac14 \times 4x = \frac14 \times -20\]

which simplifies to

\[x = -5.\]

Now try your equation.


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