… [the factor (0.03573 - 0.02781) appearing on the right-hand side] is also the part that I found to be the most complicated.
Hi Nikita. You mean the most complicated to deal with, yes? Otherwise, it's just an expression for the number 0.00792.
(Note: I'm not sure why I wrote a zero at the end of 0.00792, in my previous reply. That was unnecessary.)
When I look at that number, expressed as (0.03573 - 0.02781), I also see it as a
factor.
That is, the product is (z-2.1)/0.1 × 0.00792, and the factors are (z-2.1)/0.1 and 0.00792.
Do I divide by 0.007920 to get it on the other side? …
The phrase "get it on the other side" is not technically correct because it usually refers to adding (or subtracting) a term. Perhaps, you were thinking that division is the way to "eliminate" the factor 0.00792, on the right-hand side. If that's what you were thinking, then, yes, division could be one way to proceed. Just remember that when you divide a sum or difference by something, you must divide each term. We have
\(\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{0.1} \cdot 0.00792\)
There is one term on the left-hand side, and there are two terms on the right-hand side: 0.03573 and the
product (z-2.1)/0.1×0.00792. We divide each term:
\(\displaystyle \frac{0.035}{0.00792} = \frac{0.03573}{0.00792} - \frac{z - 2.1}{0.1} \cdot \frac{0.00792}{0.00792}\)
\(\displaystyle 4.41919 = 4.51136 - \frac{z - 2.1}{0.1}\)
To solve for z, (1) subtract 4.51136 from each side and simplify, (2) multiply each side by -0.1 and simplify and (3) add 2.1 to each side and simplify.
This thread shows that we can solve for z using different steps. Each way is valid.
For those interested in seeing yet another set of steps, here's my approach. After evaluating (2.2-2.1) and (0.03573 - 0.02781), we have
\(\displaystyle 0.035 = 0.03573 - \frac{z - 2.1}{0.1} \cdot 0.00792\)
Subtract 0.03573 from each side and simplify
\(\displaystyle -0.00073 = -\frac{z - 2.1}{0.1} \cdot 0.00792\)
The product on the right-hand side may be viewed as
\(\displaystyle -0.00073 = (z - 2.1) \cdot \frac{-0.00792}{0.1}\)
Multiply each side by (-0.1)/(0.00792) and simplify
\(\displaystyle 0.009217 = z - 2.1\)
Add 2.1 to each side and simplify
\(\displaystyle 2.109217 = z\)
A couple notes about equivalent forms (shown symbolically, below). We may rearrange the product of a ratio times a number:
\(\displaystyle \frac{A}{B} \cdot C \quad=\quad \frac{A \cdot C}{B} \quad=\quad A \cdot \frac{C}{B}\)
When a negation symbol appears in front of a ratio, we may move it to the numerator or denominator
\(\displaystyle -\frac{a}{b} \quad=\quad \frac{-a}{b} \quad=\quad \frac{a}{-b}\)
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