9 Transposing Equations
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Have you ever come across a situation during your math studies where you’re required to solve for a variable which doesn’t seem to be in the right place? Take a look at the following example to see what I mean.
[latex]\Large \text{A} = {\text{B}}^{2} \times 0.7854 \times \text{H}[/latex]
In a perfect world, you would like to solve for “A” and at the same time be given the values of both “B” and “H.”
But what if you were given “A” and you had to solve for “B”? How would you go about doing this?
The idea here would be to move the variables around and isolate “B.” What this means is that “B” is on one side of the equation by itself, and everything else is on the other side. Take a look at the equation again when this has been done.
[latex]\Large \text{B} = \sqrt{\dfrac{\text{A}}{.7854 \times \text{H}}}[/latex]
Changing the formula around is referred to as “transposing” an equation.
It’s not as simple as just moving stuff around though. There are rules to get to this point, and those rules and their application are what we are going to deal with in this part of the chapter.
The most important thing to remember when transposing equations is that whatever is done to one side of the equation must also be done to the other side of the equation.
If you look at it mathematically, this makes sense. We already determined that an equation is two mathematical expressions that are separated by an equal sign.
What this means is the addition, subtraction, division, and multiplication variables and constants on one side of the equation are equal to all the addition, subtraction, division, and multiplication variables on the other side of the equation.
So if you decide to add 5 to one side, you must add 5 to the other side. What this does is keep the equation equal.
Take a look at the following example.
[latex]\Large \begin{array}{c}10+7=9+8 \\ \text{This works out to be:} \\ 17=17\end{array}[/latex]
Here we have an equation that is true. Now add 5 to the left hand side of the equation, and you’ll see that, in order to keep the equation true, you’ll have to add 5 to the right hand side of the equation.
[latex]\Large \begin{array}{c} 10 + 7 + \mathbf{5} = 9 + 8 + \textbf{?} \\ 22 = 9 + 8 + ? \\ 22 = 9 + 8 + \mathbf{5} \\ 22 = 22 \end{array}[/latex]
Keep in mind that this is an example where we added to one side. If we had subtracted, divided, or multiplied, things would be different. We would have to do the same thing to the other side.
Transposing Equations Using Addition and Subtraction
Start with a simple equation.
[latex]\Large \begin{array}{c}7+2=8+1 \\ \text{This works out to be:}\\ 9=9\end{array}[/latex]
Then add 4 to one side of the equation and solve.
[latex]\Large 7+2+4=8+1+ \text{ ?}[/latex]
As the left hand side of the equation has had 4 added to it, the right hand side of the equation also has to have 4 added to it. We get:
[latex]\Large \begin{arr