9.3 Literal Equations
A literal equation is an equation consisting of just variables, where we solve for one of the variables. Doing so can give us new insights into the relationships between the variables.
Now that we have a sense as to what we mean by a mathematical creature of more than one variable, let's play around with some equations that have more than one variable. Until now, "solving equations" meant finding a numerical value that we could substitute back into an equation, making it a true statement. As an example, if the length of the side of a rectangle is x units while the width is 2x+1 units, and the area is 3 units, then we can write the equation:
x(2x+1)=3
and solve for numerically for x. (By the way, in this case the solution is x=.82 after using the quadratic formula and throwing away the negative solution.)
If we let A = area, l = length, w = width, then we can also write an abstract version of our equation as:
lw=A
This equation has no numbers and it's an equation of several variables. One reason that we might do this is that it gives us a different perspective as to how the three quantities in the equation relate. For example, by a little bit of algebra, we see that:
l=A/w and w=A/l.
The first relationship shows us how the length changes as we manipulate the area and the width, while the second relationship shows us how the width changes as we manipulate the area and the length.
The above example is an example of a literal equation. In literal equations, we solve for one of the variables as opposed to an actual numerical quantity. The key to solving literal equations for a particular variable is to treat your desired quantity as a variable and pretend that the other quantities are just numbers: you can then use any of the equation solving techniques that we've studied already. Literal equations can be equations of in either one or more variables. However, when equations are of several variables, the process of solving for each of them can be particularly helpful in answering a variety of different types of questions, all relating to one equation.
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If you deposit money in the bank, it typically grows using a process known as compound interest. The formula to find out how much money you have at the end of t years, if compounded annually, is given by
A=P(1+i)^t
where
A = initial amount
i = annual compounding interest rate
t = number of years
So, for example, if you invest 100 dollars at 3% interest, then after 10 years, you'll have 100(1.03)10 = $134.39.
If you're trying to save a certain amount of money (say for a new car) for some specific time in the future, and if you know how much you're going to invest, then it would be very helpful to know what interest rate you'll need to earn in order to reach your financial goal. Let's solve the literal equation A=P(1+i)^t for the interest rate:
i=
You can see the full solution below.
\begin{align*}A &= P(1+i)^t\\ \frac{A}{P} & =(1+i)^t & \textrm{Divide both sides by P}\\ \left (\frac{A}{P} \right )^{1/t}&= \left ((1+i)^t \right )^{1/t}& \textrm {Raise both sides to the 1/t}\\ \left (\frac{A}{P} \right )^{1/t}&= 1+i & \textrm{Cancel the exponents}\\ \end{align*}
Subtracting "i" from both sides we get that
i=\left (\frac{A}{P} \right )^{1/t}-1
In solving this equation we used the fact that
\left ( x^m \right )^{n}=x^mx^n
which is a relationship that we discovered about when we explored rational exponents.
We've now have one equation with many variables. The next step would be to look at many equations with many variables. What would this sort of thing look like and where could it appear? We'll take a look at this idea in the next section.
