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Lesson 1: Algebraic equations basics

# Variables, expressions, & equations

In algebra, we use variables like x, y, and z to represent unknown values. Expressions are combinations of variables and numbers, while equations equate two expressions. Variables can take on different values depending on the context, and we can evaluate expressions by substituting values for the variables. Created by Sal Khan.

## Want to join the conversation?

• What's the difference between a variable and an expression?
• A variable is an unknown value. An expression is a way of expressing a concept using an unknown value (variable). Try re-watching the video if you still don't get it.
• When you evaluate x^y for x = -2 and y = 3, you should be careful to write it as (-2)^3 and not -2^3. Remember that exponents have precedence over subtraction, so the latter is not -2 x -2 x -2, but -(2 x 2 x 2). In this case, you get -8 either way, of course.
• u should go to math heaven cuz of that. that is to hard for me
• In the example x=-2 to the power y=3 you write -2^3 that is a mistake, the right way to do it is (-2)^3 ; because the "-" sign belong to the number "2". It is no the same:
(-2)^2=4 to -2^2=-4. Thanks so much for what you are doing.
• Can I point out this is 9 YEARS ago 9 years so about 2010
• In algebra, The letter X always appears. What does X represent?
• as said above, it is used to represent an unknown number. "Solving for x" means finding which number x represents. X can actually be any letter, but x is just the most common. For example, you could also "solve for a".
• I saw the video but I still don't understand. Can someone tell the difference between expressions, equations, and inequalities in a simpler way?
• Expressions have one or more terms which are separated by plus and minus signs. All we can do with these is simplify or evaluate for given values. Examples could include x, 3x + 2y, etc.
Next, we can set two expressions equal to each other by creating an equation. This will allow us to solve or isolate a variable. Examples could be 2x + 5 = 3x - 9 or y = 3x - 2. With two variables, an equation can be a function if each input (x in the equation above) has at most one output value (y in the equation above). With a single variable, the solution is a point on the number line, and with two variables it ends up as a line or curve on a Cartesian Plane.
The inequalities (greater than, greater than or equal to, less than, less than or equal to, and not equal to) allows for multiple solutions. On a number line, it creates ray(s) or a line, and it is an area on the Cartesian Plane. The equality part of the inequality would form a line or curve which could be solid or dashed and shading either above or below this line or curve.
• is it possible that there is so many variables in a equation that the problem is unsolvable
• No. It would only make the problem more complex not unsolvable.
• Variables

In mathematics, a variable is a symbol that represents an unknown value. Variables can be used to represent anything that can change, such as length, weight, or speed. They can also be used to represent unspecified quantities, such as the number of elements in a set or the number of pages in a book.

Expressions

An expression is a combination of numbers, variables, and mathematical operations. Expressions can be used to represent any value, such as the number 5 or the expression 2 + 3.

Equations

An equation is an expression that expresses the equality of two expressions. For example, the equation x + 2 = 5 means that the value of x is 3.

Differences between variables, expressions, and equations

Here is a table that summarizes the key differences between variables, expressions, and equations:

Attribute Variable Expression Equation
Definition A symbol that represents an unknown value A combination of numbers, variables, and mathematical operations An expression that expresses the equality of two expressions
Example x 2 + 3 x + 2 = 5
Purpose To represent an unspecified quantity To represent any value To express the equality of two expressions
For example, in the equation x + 2 = 5, x is a variable that represents an unknown value. The expressions x + 2 and 5 are mathematical expressions that represent the values 7 and 5, respectively. The equation as a whole means that the value of x is 3, since 3 is the only number that makes both expressions equal.

Variables can be used in expressions and equations.
Expressions can be used in equations.
Equations can be used to solve for variables.
• I watched the video, but I'm still having a little trouble understanding what the difference is between an expression, inequality, and equation. Can someone explain it a bit simpler for me, please?
• An expression can be as simple as a number or a variable. Or, it can be a mix of numbers, variables, and math operations for addition, subtraction, multiplication and division. It will not have any equals symbol (=). Nor will it have any inequality symbol (>, <).

An equation requires 2 expression separated by an equals symbol (=).

An inequality also takes 2 expressions separated by an inequality symbol ("<", ">", "<=", ">=").

Hope this helps.
• whats the difference between an expression and an equation
• An expression can be as simple as a number or a variable. Or, it can include numbers, variables and symbols for addition, subtraction, multiplication and division. It will not have an "=" symbol or any type of inequality symbol.

An equation requires 2 expressions connect with an "=" symbol.

Hope this helps.
• 35-3× =10 how would I solve this expression
• With an equal sign, it is an equation not an expression. So subtract 35 on both sides is first opposite. This gives 35 - 35 - 3x = 10 - 35, so - 3x = -25, then divide by -3 to get -3/-3 x = -25/-3, so x = 25/3.

## Video transcript

When we're dealing with basic arithmetic, we see the concrete numbers there. We'll see 23 plus 5. We know what these numbers are right over here, and we can calculate them. It's going to be 28. We can say 2 times 7. We could say 3 divided by 4. In all of these cases, we know exactly what numbers we're dealing with. As we start entering into the algebraic world-- and you probably have seen this a little bit already-- we start dealing with the ideas of variables. And variables, there's a bunch of ways you can think about them, but they're really just values in expressions where they can change. The values in those expressions can change. For example, if I write x plus 5, this is an expression right over here. This can take on some value depending on what the value of x is. If x is equal to 1, then x plus 5, our expression right over here, is going to be equal to 1. Because now x is 1. It'll be 1 plus 5, so x plus 5 will be equal to 6. If x is equal to, I don't know, negative 7, then x plus 5 is going to be equal to-- well, now x is negative 7. It's going to be negative 7 plus 5, which is negative 2. So notice x here is a variable, and its value can change depending on the context. And this is in the context of an expression. You'll also see it in the context of an equation. It's actually important to realize the distinction between an expression and an equation. An expression is really just a statement of value, a statement of some type of quantity. So this is an expression. An expression would be something like what we saw over here, x plus 5. The value of this expression will change depending on what the value of this variable is. And you could just evaluate it for different values of x. Another expression could be something like, I don't know, y plus z. Now everything is a variable. If y is 1 and z is 2, it's going to be 1 plus 2. If y is 0 and z is negative 1, it's going to be 0 plus negative 1. These can all be evaluated, and they'll essentially give you a value depending on the values of each of these variables that make up the expression. An equation, you're essentially setting expressions to be equal to each other. That's why they're called equations. You're equating two things. An equation, you'll see one expression being equal to another expression. For example, you could say something like x plus 3 is equal to 1. And in this situation where you have one equation with only one unknown, you can actually figure out what x needs to be in this scenario. And you might even do it in your head. What plus 3 is equal to 1? Well, you could do that in your head. If I have negative 2, plus 3 is equal to 1. In this context, an equation is starting to constrain what value this variable can take on. But it doesn't have to necessarily constrain it as much. You could have something like x plus y plus z is equal to 5. Now you have this expression is equal to this other expression. 5 is really just an expression right over here. And there are some constraints. If someone tells you what y and z is, then you're going to get an x. If someone tells you what x and y is, then that constrains what z is. But it depends on what the different things are. For example, if we said y is equal to 3 and z is equal to 2, then what would be x in that situation? If y is equal to 3 and z is equal to 2, then you're going to have the left-hand expression is going to be x plus 3 plus 2. It's going to be x plus 5. This part right over here is just going to be 5. x plus 5 is equal to 5. And so what plus 5 is equal to 5? Well, now we're constraining that x would have to be equal to 0. Hopefully you realize the difference between expression and equation. In an equation, essentially you're equating two expressions. The important thing to take away from here is that a variable can take on different values depending on the context of the problem. And to hit the point home, let's just evaluate a bunch of expressions when the variables have different values. For example, if we had the expression x to the y power, if x is equal to 5 and y is equal to 2, then our expression here is going to evaluate to, well, x is now going to be 5. y is going to be 2. It's going to be 5 to the second power, or it's going to evaluate to 25. If we changed the values-- let me do that in that same color-- if we said x is equal to negative 2 and y is equal to 3, then this expression would evaluate to-- let me do it in that-- negative 2. That's what we're going to substitute for x now in this context. And y is now 3, negative 2 to the third power, which is negative 2 times negative 2 times negative 2, which is negative 8. Negative 2 times negative 2 is positive 4, times negative 2 again is equal to negative 8. We could do even more complex things. We could have an expression like the square root of x plus y and then minus x, like that. Let's say that x is equal to 1 and y is equal to 8. Then this expression would evaluate to, well, every time we see an x, we want to put a 1 there. So we would have a 1 there, and you'd have a 1 over there. And every time you see a y, you would put an 8 in its place in this context. We're setting these variables. So you'd see an 8. Under the radical sign, you would have a 1 plus 8. So you'd have the principal root of 9, which is 3. This whole thing would simplify. In this context, when we set these variables to be these things, this whole thing would simplify to be 3. 1 plus 8 is 9. The principal root of that is 3. And then you'd have 3 minus 1, which is equal to 2.