Algebraic Substitution for Electric Circuits

Substitution is a technique whereby we let a variable represent (stand in the place of) another variable or an expression made of other variables. One application where we might use substitution is when we must manipulate an algebraic expression containing a lot of similar-looking variables, as is often the case with science problems.

Take this series-parallel resistor circuit for example:

The equation expressing total resistance as a function of the four resistor values looks like this:

$$R_{total} = R_1 + \frac{R_2 (R_3 + R_4)}{R_2 + R_3 +R_4}$$

Now imagine being asked to manipulate this equation to solve for R₃. When the only visual feature distinguishing each of the variables is the subscript (total, 1, 2, 3, or 4), it becomes very easy to lose track of where one is in the algebraic manipulation. A very common mistake is to exchange or needlessly repeat subscripts during the process, effectively mis-placing one or more variables. To help avoid such mistakes, you may substitute different letter variables for R_total, R₁, R₂, R₃, and R₄ like this:

Original variable New variable Rtotal y R1 a R2 b R3 c R4 d

$$y = a + \frac{b(c+d)}{b + c + d}$$

After doing the algebraic manipulation to solve for c (R₃), the equation looks like this:

$$c = \frac{(y-a)(b+d)-bd}{a+b-y}$$

Back-substitute the original R variables in place of a, b, c, d, and y as you see them in the above equation to arrive at a form that directly relates to the schematic diagram.

Substitution is a technique whereby we let a variable represent (stand in the place of) another variable or an expression made of other variables. One application where we might use substitution is when we must manipulate an algebraic expression containing multiple instances of the same sub-expression. For example, suppose we needed to manipulate this equation to solve for c:

$$1 = \frac {a+b(d^2-f^2)+c}{d^2-f^2}$$

The sub-expression d² − f² appears twice in this equation. Wouldn’t it be nice if we had something simpler to put in its place during the time we were busy manipulating the equation, if for no other reason than to have less variables to write on our paper while showing all the steps to our work? Well, we can do this!

Substitute the variable x for the sub-expression d² − f², and then solve for c. When you are done manipulating the equation, back-substitute d² − f² in place of x.

Substitution is the term we give to the mathematical equivalence of one variable to one or more other variables in an expression. It is a fundamental principle used to combine two or more equations into a single equation (among other things).

For example, we know that the formula for calculating current in a simple one-resistor circuit is as follows:

We also know that the total resistance (R) of a three-resistor series circuit is as follows:

Combine these two equations together using substitution so that we have a single equation for calculating current I in a three-resistor series circuit given the source voltage V and each resistance value R₁, R₂, and R₃:

In other words, you need to have as your answer a single equation that begins with “I = ” and has all the variables V, R₁, R₂, and R₃ on the other side of the ëqual” sign.