Algebra and Arithmetic
First, you need to know that algebra and arithmetic have many similarities. It follows all the rules of arithmetic and uses the four basic operations upon which arithmetic is built: addition, subtraction, multiplication, and division. However, algebra introduces a new element… the unknown.
Basic Algebraic Equations
You might encounter a problem like this: 4 + 6 =? Before learning algebra, we would write the problem like this: 4 + 6 = "blank," and then fill in the answer after calculating it.
But in algebra, we would write it like this: 4 + 6 = x. An important concept in algebra is that when we don't know the value of a number, we use a symbol to represent it. This symbol is usually any letter from the alphabet. The letter "x" is a very commonly used symbol. Here, "x" is a placeholder representing the number we don't yet know. This is a very basic algebraic equation.
An equation is a mathematical expression that states that two things are equal. In this example, our equation tells us that the known value on one side of the equals sign (4 + 6) is equal to the value on the other side. This unknown value happens to be what we call "x." One of the main goals of algebra is to find the unknown in an equation. Doing this is called "solving the equation." You simply add 4 and 6 on one side of the equation, which becomes 10 = x, or x = 10. So now we know what "x" is 10.
Complex Equations
Doesn't that seem too easy? So in algebra, you'll usually encounter more complex equations like this: x - 5 + 2 = 1. This is essentially the same as x = 5 - 2 + 1, just in a different form, so it's not easy to see the value of x directly.
Therefore, in algebra, solving equations is like playing a game where you're faced with a bunch of complex equations, and your task is to simplify and rearrange them until you get a simple and clear equation (like x = 5 - 2 + 1).
Rules of Signs
Before we learn how to solve more complex equations, let's learn some important rules about the use of signs in algebraic equations.
1. The same symbol represents the same value in the same problem.
The first rule you need to know is that the same symbol (or letter) can represent different unknown numbers in different algebra problems. For example, in the problem we just solved, the letter "x" represented the number 4, right?
However, in different problems, "x" can represent different numbers. For example, what if someone asked us to solve the equation 5 + x = 10? For this equation to be true, "x" must be equal to "5" in this equation, because 5 + 5 = 10. So, "x" (or any other symbol) can represent different values in different problems. That's fine, but a symbol cannot represent different values simultaneously in the same problem! For example, what if the equation was x + x = 10?
This equation means that x plus x equals 10. There are many different numbers that add up to 10, such as 6 plus 4. But if the first x represented 6 and the second x represented 4, then x would be representing two different values at the same time, which would be very confusing!
2. Different symbols can represent the same value, or they can represent different values.
If you want symbols to represent two different numbers at the same time, you need to use two different symbols, such as x and y. Suppose you have the equation: x + y = 2. What would x and y have to be for the equation to be true? If x = 0 and y = 2, then the equation is true. Or, we could swap them. If x = 2 and y = 0, the equation is also true. But there's another possibility: if x = 1 and y = 1, then the equation is also true, right?
Variables
Did you notice that the equation x + y = 2 has different solutions? In other words, the value of "x" can be 0, 1, or 2, depending on the value of "y". If "x" is 0, then "y" must be 2. In algebra, x and y are both variables because their values change depending on each other's values. It's common in algebra to use letters to represent variables because letters can represent different values in different problems.
"Default" operation
Multiplication is the "default" operation. This means that if no other arithmetic operation is shown between two symbols, you can assume they are being multiplied. The multiplication operation is "implied." For example, instead of writing "a" multiplied by "b," you can omit the multiplication sign and simply write "ab." The advantage of this multiplication rule is that it makes many algebraic equations more concise and easier to write. For example, instead of writing a * b + c * d = 10, you can simply write ab + cd = 10.
Applications of Algebra
Is it useful in the "real world"? Or is it just a bunch of tricky problems to keep students busy in school? In fact, algebra is very useful in describing things in the real world.
For example, there is a type of equation in algebra called a "linear equation" because its graph is a straight line. These equations can help you describe the slope of a roof or tell you how long it will take to get to a certain place. Another type of algebraic equation, called a "quadratic equation," can be used to design the lenses of a telescope or describe the trajectory of a ball in flight.
Summary
Algebra extends arithmetic by introducing the use of unknowns (usually letters, such as "x"). If you find algebra difficult right now, that's okay! Socratic AI (ScanMath)'s free algebra solver can guide you through tricky problems and help you build confidence. Everyone starts from scratch, and the more you practice, the easier it will become.
