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Solve Formulas for a Given Variable

Solving equations in terms of other variables is very similar to solving any other equation. You need to manipulate the equation to isolate the variable you are trying to solve for. You won't get a constant answer such as $x=3$ or $y=\frac12$. Instead, you'll get one variable defined by another variable, such as $x=2y$.

Example:

Solve $2x+3y=6$ for $y$.

Since we're solving for $y$, we want to get the $y$ variable alone on one side of the equation. Let's look at it step-by-step. $$\begin{align} 2x+3y&=6\\ 2x\,{\color{#8925AE}{-\,2x}}+3y&=6\,\color{#8925AE}{-\,2x}\\ 3y&=6-2x \end{align}$$

Now that the $y$ variable is alone on one side, all we have to do is isolate it completely. $$\begin{align} 3y&=6-2x\\ \frac{3y}{\color{#8925AE}{3}}&=\frac{6-2x}{\color{#8925AE}{3}}\\ y&=2-\frac23x \end{align}$$

We can see that this new equation is actually equivalent to our original equation, which we can do because of the properties of equality, the multiplication & division property of equality and the addition & subtraction property of equality!

Let's look at one more example with some images to walk us through. Remember, when looking at models, the grey line represents the equal sign.

Solve $5x-y=10$ for $y$.

If you solve for your variable, $x$, you can check your solution by substituting the solved $x$ value into both equations. You can learn more about that in a later lesson!

Most literal equations have more than one variable, so you'll be asked to solve the equation in terms of one of the variables. For instance, if the literal equation contains variables $x$ and $y$, you might just be asked to solve for $x$ in terms of $y$.

This explanation includes:

• Solving for a given variable $x$ in terms of other variable(s)
• Solving for a given variable $x$ (and getting a numerical answer)

Solving for a given variable $x$ in terms of other variable(s)

Let's start with a pretty basic literal equation problem. First, we'll create a literal equation from this word problem:

Darryl is on his way to buy chicken and ribs for his family's cookout. He is going to spend exactly $191$ dollars (he has a big family!). Darryl knows that each chicken costs $7$ dollars and each pack of ribs costs $\text{\textdollar}19.50$. Come up with an equation that models this information.

Unless the problem instructs you otherwise, you can choose whatever letters you want as your variables. We're just going to stick with the standard $x$ and $y$ here, but you could certainly choose something like $c$ (for chicken) and $r$ (for ribs).

$x$ is the number of chickens, and each chicken costs $\text{\textdollar}7$, so $7x$ represents how much money Darryl will spend on chickens in total. Using that logic, we can write the total cost of all the ribs as $19.50y$, and then set those two together equal to the budget of $\text{\textdollar}191$:

$$7x + 19.50y = 191$$

Whenever you see "[something] in terms of [other thing]," that signals that the [something] should be on one side of the equation, with everything else on the other side.

As messy as it looks, that last step shows your answer—you've solved for $x$ in terms of $y$. Or, in other words, you've solved for the number of chickens in terms of the number of ribs.

Solving for a given variable $x$ (and getting a numerical answer)

Let's stick with the same literal equation, but now solve for one of the variables so that we get a numerical answer. Just as a refresher, here is our literal equation, where $x$ is the number of chickens and $y$ is the number of packs of ribs:

$$7x + 19.50y = 191$$

Consider this question now:

Darryl has decided that he needs to buy $8$ packs of ribs. Determine how many chickens he will be able to buy.

When you're given a value for one of the variables, plug it into your literal equation. Then, solve for the other variable.

In this case, we know that Darryl wants to buy $8$ packs of ribs, so we're going to plug in $8$ for $y$.

Let's try something a little messier now.

Darryl is considering buying only $5$ packs of ribs. How many chickens will he be able to purchase now?

We can plug $5$ into our equation for $y$ to solve for the new number of chickens: $7x+19.50(5)=191$.

Darryl has enough money to afford as many as $13$ whole chickens, but not enough to buy $14$ chickens. So, we can say that Darryl can buy $13$ chickens if he buys $5$ packs of ribs.

(And then maybe Darryl can invite us to his cookout!)

(Video) Solving Literal Equations

by YourMathGal

This video YourMathGirl explains what literal equations. We can use this video to help us solve literal equations in terms of one of the variables.

$\large{\text{Summary}}$

A literal equation is an equation with more than one variable.

The equation for the volume of a box, $V=lwh$, is a literal equation because it has four variables that are unknown. If you wanted to solve for a certain variable, like $l$, you would divide both sides by $w$ and $h$. This would give $l=\frac{V}{wh}$.

The trick is to pretend the other variables are numbers and apply any operation that you would normally apply in order to move the variables around in the equation. Just always remember: whatever you do to one side of the equation, you have to do to the other!

To solve for $r$, we need to get $r$ all alone on one side of the equation. We can do this by just using a couple of operations on both sides of the equation so that terms move from one side to the other.

$$S = 4\pi r^2$$

Right now, $r$ is being multiplied by $4$ and $\pi$, and squared. First, let's get $r^2$ alone on one side of the equation by dividing both sides by $4\pi$. $$\begin{align} S &= 4\pi r^2\\ \frac{S}{\color{#8925AE}{4\pi}} &= \frac{4\pi r^2}{\color{#8925AE}{4\pi}}\\ \frac{S}{4\pi} &= r^2 \end{align}$$ Alright! Now we just have to make it so that $r$ isn't squared anymore by taking the square root of both sides. $$\begin{align} \frac{S}{4\pi} &= r^2\\ \color{#8925AE}{\sqrt{\color{#8925AE}{\frac{S}{4\pi}}}} &= \color{#8925AE}{\sqrt{\color{#8925AE}{r^2}}}\\ \sqrt{\frac{S}{4\pi}} &= r \end{align}$$

And then, to make it look like the equations in the answers up there, we can just swap things around the equals sign due to the symmetric property of equality. $$r = \sqrt{\frac{S}{4\pi}}$$