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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

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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.

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.

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.

So

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

and

2*y* = 2(*x* + 5) or 2*y* = 2*x* + 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 - 3*x*

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 - 3*x* - 3

Which simplifies to

*x* = 2 - 3*x*

Next, we add 3*x* 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* + 3*x* = 2 - 3*x* + 3*x*

Which simplifies to

4*x* = 2

Because 4*x* 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*.

4*x*/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 - 3*x*

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

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

2*x* + *y* = 4

4*y* = 2*x* + 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.

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.

So, let's isolate the *y* in the top equation.

2*x* + *y* - 2*x* = 4 - 2*x*

*y* = 4 - 2*x*

Because we now know that *y* is equal to 4 - 2*x*, we can substitute the 4 - 2*x* in for *y* in the other equation. Now that equation will look like this.

4(4 - 2*x*) = 2*x* + 6

Now we can just solve this equation for *x*. The first step is to subtract 2*x* from both sides, which gives us

16 - 10*x* = 6

Then we subtract 16 from both sides to get

-10*x* = -10

Dividing by 10 gives us

*x* = 1

Because we know that *x* is equal to 1, we can easily solve for *y*. Again, it doesn't matter which equation you use.

*y* = 4 - 2*x*

Substitute the 1 in for *x* and we get

*y* = 4 - 2(1)

Which simplifies to

*y* = 4 - 2 or *y* = 2

Now we have solved the equation and we know that

*x* = 1 and *y* = 2

Let's try the same example using the other method for solving two variable equations. For this method, the goal is to be able to add our equations together and, in the process, have one of the variables be eliminated. The only way this will work is if, when you add, one of the variables has a sum of zero.

The example will show what I mean. And remember, you can do whatever you want to any equation, as long as you do the same thing to both sides of the equation.

Remember that our example equations are

2*x* + *y* = 4

4*y* = 2*x* + 6

The first thing I notice is that the *x* variables have the same **coefficient**, or number in front of the variable, but they are on opposite sides of their equations. So my first idea is to subtract 2*x* from both sides of the second equation.

When we subtract 2*x* from both sides of

4*y* = 2*x* + 6

We get

4*y* - 2*x* = 6

Now we can add the two equations together and the *x* variable will be eliminated because 2 + (-2) is zero.

When we add the two equations together, we get

5*y* = 10

This is an easy problem to solve. Just divide both sides of the equation by 5 and you will know what *y* is equal to.

5*y*/5 = 10/5

Which means that

*y* = 2

The final step is just a matter of substituting 2 in for *y* in one of the above equations to determine that *x* is equal to 1.

Manipulating functions can be accomplished by remembering one simple rule. In order to keep the equations balanced, you must do the same thing to each side of the equation. Whether it is dividing by 3 or adding 7, as long as you do the same thing to both sides of the equation, it will remain equal, or balanced.

The same rule applies when you are trying to solve equations with more than one variable. There are two methods for solving these types of equations. The first is to isolate one of the variables in one of the equations and then substitute it in the second equation. The second method involves adding both equations together in order to eliminate one of the variables.

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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

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