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Fundamentals of Algebra

Algebra is a framework for solving problems. It gives us the tools to answer questions that follow certain patterns. You just have to recognize the pattern and then apply the appropriate rules to get the solution you need. That is the power of algebra.

Variables and Expressions

Our main algebraic tool is the expression. Expressions are mathematical statements. We can write them in either English words or mathematical symbols. As long as the statement expresses some mathematical concept, it is an expressions.

Every expression contains a combination of constants, variables, coefficients and operators.

Constants are pure numbers. The number 2 is a constant. Constants may be variable themselves, but the must remain as they are through all algebraic operations. You can recognize Constance because they are numbers or letters sitting by themselves.

Variables are placeholders. They represent numbers and other expressions. Variables are our main algebraic tools. We use them to showcase generic rules that apply to all numbers of a given set without exception.

We use letters to write variables. We can use any letter we want as long as our variables are consistent, or mean the same thing throughout our work. There are some conventions you can use for some common situations.

  1. We use an x for a general unknown number. We sometimes use y or z for this as well.
  2. We use n for only natural numbers
  3. We use z for complex numbers
  4. We use u for general unknown expressions.

Coefficients are variables or constants multiplied to our main variables.  Our main variables will appear in multiple terms (or sub-expressions) in our expression. Though, any variable can serve as a coefficient for another variable. We usually just call our main variables “variables” and our coefficient variables “coefficients”.

Operators are symbols that represent the operators we doing to our variables and constants. For instance, the plus sign, +, is the operator for addition.

The equals sign, =, is the operator for equality. It sets the two sub-expressions on its sides as equal values. We call any expression that contains an equals sign equations.

Operators split expressions into terms. Terms are any combination of numbers ad variables that represent an applied multiplication or division.

For example, in the following expression

  • The variable x is the variable.
  • The variables a, b, and c are the coefficients.
  • The symbols + and = are operators.
  • The variable c is a constant.

Solving Equations

Our main goal for solving equations is to clear the variable of any constants or coefficients. Thus, we will end up with the variable on one side of the equation and a constant on the other. This new constant term is our solution. However, we must do it so that we maintain the equality across the equal sign.

We use the exact opposite operation than the operator attached each constant term and coefficient. The trick is to do the same operation on each side of the equal sign. If the terms are link to the variable through addition, we use subtraction. If the existing operation is subtraction, we use addition. Just make sure we add or subtract a constant equal to the constant or coefficient we want to remove from our variable. This changes our equation to a simpler one without ruining the equality. Once we only have the variable on one side of the equation and everything else on the other, we have solved the equation.

Solving Equations Using Addition and Subtraction

We clear constant terms by adding or subtracting them away. We use the exact operation from the one already present. We add the constant when the existing operation is a subtraction. We subtract the constant when the operation is addition.

For example, in the following equation, we subtract the constant c from both sides of the equation to clear the variable x.

Remember that subtraction is just adding a negative.

Doing the same operation on both sides of the equation keeps the equation balanced. This is the only way to ensure that our result is both correct and makes sense. You can check you answer by substituting our solution constant for the variable in our original equation to see if both sides of the equation end up as the same constant.

We use a similar procedure to deal with coefficients. This time we use division instead. That is if we divide both sides of our equation by the coefficient, we will remove that coefficient from the variable. We usually use this procedure we have no more constant terms to subtract away.

Solving Equations with Variables on Both Sides

This procedure lets us deal with problems with the variable on both sides of the equation. In this situations, we use whatever procedure helps us move the variable to one side of the equation. It does not matter which side we choose. We just have to choose one.

We want to end up with having only one term with the variable.

Linear terms are the easiest to deal with. We just add or subtract them until all linear terms (or terms that contain only the variable and its coefficient) are on one side. We then factor out the variable from all like terms and add the coefficients together. We then divide the confidents to the other side of the equals.

Non-linear terms require more work, and may require additional divisions or some other step.

Change of Variables

Let’s go back to the note about using the variable u for unknown expressions. While reading lists of algebraic expressions, you will notice how much they use u as the variable instead of x. This is because you can use the procedure called change of variable to change any expression into another simpler expression when solving algebra problems.

Change of variables works great with complex problems that would be difficult or impossible to do otherwise. With it, you look for any sub-expression that seems to act like a variable. You usually find them in factor terms, trigonometry, and other functions of functions. Once you spot an appropriate expression, you set it equal to an alias variable, such as u, and substitute this alias as the expression in the problem. You then solve the problem for the alias. Once there, you undo the substitution, and solve for your original variable.

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