Answer:
Any of ...
- add 5 to both sides (probably the best choice)
- multiply both sides by 3
- multiply both sides by 3/2
Explanation:
You want to know the first step in solving the equation ...
2/3x -5 = 27
Steps
This can be considered to be a "2-step" linear equation. It has a variable term and a constant term on one side of the equal sign, and a constant on the other side.
Separate variables from constants
The usual first step in solving a "2-step" linear equation is to add the opposite of the constant that is not by itself. This puts the variable term by itself on one side of the equation, and a constant on the other side.
If the first step is "add 5", the result would be ...
2/3x = 32
Eliminate fractions
When an equation involves fractions, often the first step is clear fractions. This is done by multiplying the equation by a common denominator (usually the least common denominator). Here, the denominator of the only fraction is 3.
If the first step is "clear fractions", the result would be ...
2x -15 = 81
X-coefficient
Recognizing that we ultimately want x by itself, there is another choice of first step. We can choose as a first step make the coefficient of x be 1. This requires multiplying the equation by 3/2.
If the first step is "clear the x-coefficient", the result would be ...
x -15/2 = 81/2
We notice this leaves us some fractions to deal with, so it might not be the best choice of first step.
Best choice
The "best choice" of first step will be the one that matches the expectation of your grader. The step of "add 5" is very attractive in that it involves integer arithmetic and results in a number that works nicely with the subsequent operation of multiplying by 3/2.
The "clear fractions" step makes the "2-step" equation into a "3-step" equation by adding an unnecessary step. (If you're uncomfortable with arithmetic involving fractions, that may be a cost you're willing to pay.)
If the integers in the equation were even, then multiplying by 3/2 as a first step would also be attractive. As it is, the result is somewhat messy, though the fractions do have the same denominator. This makes them relatively easy to work with.
Our conclusion is that the first step could be ...
Any of ...
- add 5 to both sides (probably the best choice)
- multiply both sides by 3
- multiply both sides by 3/2
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Additional comment
There's almost always more than one path to the solution of a math problem. Though a first step is required, what that step is may depend on the path you want to take.
One could even argue that the "first step" is analyze the problem to see what you are given, and what you are expected to find.
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