Manipulating Functions and Solving Equations for Different Variables

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  • 0:05 Braving the Unknown
  • 0:31 Manipulating Functions
  • 4:04 Solving Equations for…
  • 8:57 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
Functions can be manipulated to solve for many different variables. There are certain rules to follow, but if the rules are adhered to, solving equations can be quite simple. This lesson will show you how.

Braving the Unknown

If you are like most people, you like things to be consistent and the same. You don't like change and fear the unknown. Fortunately, there are times when we just have to face our fears, step out and brave the unknown.

Mathematics is full of unknowns. And they can be downright scary. But by following the rules and not panicking, you can brave the unknowns. You can manipulate and solve functions, even if they have more than one variable.

Manipulating Functions

Functions work like a scale, with the equal sign working as the fulcrum, or pivot point on the scale. With a scale, or balance, you can add weight to either side of the scale, and as long as the weights are equal, the scale remains balanced.

This is also true with a function. If you add (or subtract, multiply or divide) something to one side of the equation, you have to do the same to the other side in order to keep the equation balanced.

For example, look at this function

y = x + 5

You can manipulate this equation in many different ways, and as long as you perform the same operation to each side of the equal sign your equation remains equal.


y - 2 = x + 5 - 2 or y - 2 = x + 3


2y = 2(x + 5) or 2y = 2x + 10

You can use this rule to manipulate functions in a variety of ways. The most common way is to solve the equation, which means to manipulate the equation to isolate a variable.

Check out this example:

Solve for x

x + 3 = 5 - 3x

The only way to solve this equation is by manipulation. We need to have all of the numbers on one side and the variables (in this case, x) on the other side. Which side you choose for each does not matter.

The first step is to move all the numbers to one side of the equation. It is most common to move them to the right side, so that's what we'll do. The way to do this is to perform the opposite operation on any numbers on the left hand side of the equal sign. For this example, that means that we will subtract 3 from each side of the equation

x + 3 - 3 = 5 - 3x - 3

Which simplifies to

x = 2 - 3x

Next, we add 3x to each side of the equation to 'move' the variable to the left side of the equation. Remember, to keep the equation equal, we need to perform the same operation to each side of the equation.

x + 3x = 2 - 3x + 3x

Which simplifies to

4x = 2

Because 4x really is 4 * x, the next step is to divide each side by 4. This will isolate the variable on the left side of the equation and put all the numbers to the right side, solving the equation for x.

4x/4 = 2/4

Which simplifies to

x = ½

To check our answer, we can substitute ½ for x and see if both sides of the equation remain equal.

x + 3 = 5 - 3x

Substituting in 1/2 for x, we get

1/2 + 3 = 5 - 3(1/2)

Which simplifies to

3 ½ = 5 - 1 ½

3 ½ = 3 ½

Since the equation is true, we know we obtained the correct answer and x = ½

Solving Equations for Different Variables

The rules for solving equations and manipulating formulas still apply when the variables are different. They also apply when there is more than one variable in a formula or set of formulas.

In order to solve equations with more than one variable, you must have more than one equation.

Let's try this example

2x + y = 4

4y = 2x + 6

There are multiple methods for solving equations with two variables. I will demonstrate two of them and let you decide which you prefer. They both will give the correct answer.

The first method for solving an equation with two variables is substitution. Here are the steps.

1. Manipulate one equation so that one of the variables is alone on the left side of the equation.

This can be done to either equation, although it's best to pick the one that will give you the least amount of work. In this example, I would choose to get the y alone in the top equation because that will only involve one step. Isolating the y in the second equation will also only take one step, but in looking at it, I see that dividing everything by 4 will give me fractions on the right side of that equation, which are always more difficult to work with. If the task of choosing seems daunting, just relax. If you are careful, it won't matter which equation you start with. And, with practice, you will learn to quickly see what equation will be easier to start with.

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