Formulas are equations that contain more than one variable. And, there's a couple of kind of problems
Formulas are equations that contain more than one variable. And, there's a couple of kind of problems

that we want to work involving formulas. Here's our first one, this is a formula: 2x + 3y = 6. Let's find y if x = 3.
that we want to work involving formulas. Here's our first one, this is a formula: 2x + 3y = 6. Let's find y if x = 3.

So, that's the first type of problem, where we substitute for all of the variables except one, and then solve for that variable.
So, that's the first type of problem, where we substitute for all of the variables except one, and then solve for that variable.

So, I'm going to take this equation right here, this formula, and I'm going to substitute in 3 for x. If I do that, I have 2(3) + 3y = 6.
So, I'm going to take this equation right here, this formula, and I'm going to substitute in 3 for x. If I do that, I have 2(3) + 3y = 6.

2(3) = 6, + 3y =6. I'll add -6 to both sides, and end up with 3y = 0.
2(3) = 6, + 3y =6. I'll add -6 to both sides, and end up with 3y = 0.

Multiply both sides by 1/3, and I end up with y = 0. So, y = 0 when x = 3 if 2x + 3y = 6.
Multiply both sides by 1/3, and I end up with y = 0. So, y = 0 when x = 3 if 2x + 3y = 6.

So, given this formula right here, we can find any value of y for any given value of x,
So, given this formula right here, we can find any value of y for any given value of x,

simply by substituting and then solving the linear equation that results.
simply by substituting and then solving the linear equation that results.

The second kind of problem that we want to work involving formulas is to solve the formula for one of the variables,
The second kind of problem that we want to work involving formulas is to solve the formula for one of the variables,

without being given replacements for the others. So, for instance, here's the formula for the area of a rectangle: A = (l)(w);
without being given replacements for the others. So, for instance, here's the formula for the area of a rectangle: A = (l)(w);

area is equal to length times width. Let's solve this for the variable l. So when I do that, I consider this to be the variable,
area is equal to length times width. Let's solve this for the variable l. So when I do that, I consider this to be the variable,

and everything else to be just a constant, like a number. So, I have A = (l)(w). I want to solve for l, so I'm going to divide the right side by w.
and everything else to be just a constant, like a number. So, I have A = (l)(w). I want to solve for l, so I'm going to divide the right side by w.

That will give me just l over here. If I do that to one side, I have to do it to the other side also. So A ÷ w = l.
That will give me just l over here. If I do that to one side, I have to do it to the other side also. So A ÷ w = l.

And it's usually best if we reverse sides and write this as l = A/w, like that. So, A = (l)(w), and l = A ÷ w.
And it's usually best if we reverse sides and write this as l = A/w, like that. So, A = (l)(w), and l = A ÷ w.

Both say the same thing about the relationship between those three variables A, l, and w.
Both say the same thing about the relationship between those three variables A, l, and w.

This one is solved for A. This one is solved for l. Let's try another one: how about if we have P = a + b + c, and we want to solve for a.
This one is solved for A. This one is solved for l. Let's try another one: how about if we have P = a + b + c, and we want to solve for a.

Well, I'll start by adding -b to both sides and -c to both sides. So, I have P +
Well, I'll start by adding -b to both sides and -c to both sides. So, I have P +

P + (-b) +(-c) = -a + b + (-b) + c + (-c). So, I've added -b to both sides;
P + (-b) +(-c) = -a + b + (-b) + c + (-c). So, I've added -b to both sides;

and I added -c to both sides. On the left side, let's change back to subtraction:
and I added -c to both sides. On the left side, let's change back to subtraction:

P - b - c = a + 0 + 0, which is just a. So, now let's rewrite it this way: a = P - b - c.
P - b - c = a + 0 + 0, which is just a. So, now let's rewrite it this way: a = P - b - c.

So, there's the exact same formula for the four variables: a, b, c, and P. This one is solved for P.
So, there's the exact same formula for the four variables: a, b, c, and P. This one is solved for P.

This one is solved for a. What do I use to solve these formulas? Just the addition property of equality and the multiplication property.
This one is solved for a. What do I use to solve these formulas? Just the addition property of equality and the multiplication property.

Let's try one more. Solve for y: 6x + 3y = 12. I want to isolate y on one side of the equation;
Let's try one more. Solve for y: 6x + 3y = 12. I want to isolate y on one side of the equation;

I'll add -6x to both sides. When I do, I have 3y = -6x + 12 adding -6x to this side,
I'll add -6x to both sides. When I do, I have 3y = -6x + 12 adding -6x to this side,

and -6x to this side. Next, I'm going to multiply both sides by 1/3. 1/3(3y) = 1/3 times this whole side right here, so 6x + 12.
and -6x to this side. Next, I'm going to multiply both sides by 1/3. 1/3(3y) = 1/3 times this whole side right here, so 6x + 12.

I need to use those parentheses, so I'm multiplying 1/3 times this whole side of the equation. 1/3(3y) = y;
I need to use those parentheses, so I'm multiplying 1/3 times this whole side of the equation. 1/3(3y) = y;

1/3(-6x) = -2x; 1/3(12) = 4. So, this formula right here,
1/3(-6x) = -2x; 1/3(12) = 4. So, this formula right here,

y = -2x + 4, says the same thing about x and y that this formula does.
y = -2x + 4, says the same thing about x and y that this formula does.

This one doesn't have any of the variables solved for it. This one is solved for y. So, there's a look at how we work with formulas in mathematics.
This one doesn't have any of the variables solved for it. This one is solved for y. So, there's a look at how we work with formulas in mathematics.