2. Solve for   x   and check.

The following steps are suggested for solving an equation.

  1. Clear denominators. Multiply both sides by a common denominator.
  2. Simplify. Remove parentheses and combine like terms.
  3. Transpose known terms to one side and unknonwn terms to the other.
  4. Combine.
  5. Divide both sides by the coefficient of the unknown.
  6. Check.

We say that these steps are suggested only because there may be equations where one would want to proceed differently. If that is the case, you will probably find that the steps used are contained in this list, but they may be used in a different order. This order will work best in the most number of cases. Fractions are caused by division. If we clear denominators, we will have polyomials which are simpler than fractions, so step 1 is a good first step. At that point, if there are parenthesses to remove of like terms to combine, do so. At this point we can tell with what kind of an equation we are working. The steps for solving polynomial equations differ with the various possible degrees. These steps are for equations which turn out tobe first degree after clearing denominators. If we have first degree polynomials, after removing parentheses and combining like terms, they will simplify down to at most an unknown term and a known term on both sides. So transposing known terms to one kise and unknown terms to the other will involve transposing at most two terms. After combining, we will have a single unknown term on one side and a single known term on the other. This will put all of our unknowns together in one place with only one thing happening to it. It will probably be being myltiplied by some kind of a coefficient. Get rid of that, by dividing by that coefficient, and we will have the equation solved for the unknown. The checks are a great deal of fun. It is very satisfying to see that the solution will make both sides of the equation equal.

In this case, clearing denominators amounts to cross multiplying.

2(5x-7) = 2x+3

Remove parentheses.

10x - 14 = 2x + 3

Transpose known terms to one side and unknown terms to the other.

10x - 2x = 14 + 3

Combine.

8x = 17

Divide both sides by the coefficient of the unknown.

Check: Copy down the original equation

except where there is an x, copy down the solution in parentheses.

This gives us an interesting order of operations problem. Perform the multiplications in the bottoms first.

Next are the addition and subtraction in the bottoms.

For this we will need common denominators.

We can now invert and multiply.

which checks.

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