Algebra
Elementary algebra

Elementary algebra, also called school algebra, college algebra, and classical algebra, is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on variables and examines how mathematical statements may be transformed.

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition, subtraction, multiplication, division, exponentiation, extraction of roots, and logarithm. For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in 2 + 5 = 7 {\displaystyle 2+5=7}.

Elementary algebra relies on the same operations while allowing variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the equation 2 × 3 = 3 × 2 {\displaystyle 2\times 3=3\times 2} belongs to arithmetic and expresses an equality only for these specific numbers. By replacing the numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the commutative property of multiplication, which is expressed in the equation a × b = b × a {\displaystyle a\times b=b\times a}.

Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention, the lowercase letters x {\displaystyle x}, y {\displaystyle y}, and z {\displaystyle z} represent variables. In some cases, subscripts are added to distinguish variables, as in x 1 {\displaystyle x_{1}}, x 2 {\displaystyle x_{2}}, and x 3 {\displaystyle x_{3}}. The lowercase letters a {\displaystyle a}, b {\displaystyle b}, and c {\displaystyle c} are usually used for constants and coefficients. The expression 5 x + 3 {\displaystyle 5x+3} is an algebraic expression created by multiplying the number 5 with the variable x {\displaystyle x} and adding the number 3 to the result. Other examples of algebraic expressions are 32 x y z {\displaystyle 32xyz} and 64 x 1 2 + 7 x 2 − c {\displaystyle 64x_{1}^{2}+7x_{2}-c}.

Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is a statement formed by comparing two expressions, saying that they are equal. This can be expressed using the equals sign (= {\displaystyle =}), as in 5 x 2 + 6 x = 3 y + 4 {\displaystyle 5x^{2}+6x=3y+4}. Inequations involve a different type of comparison, saying that the two sides are different. This can be expressed using symbols like the less-than sign (< {\displaystyle <}), the greater-than sign (> {\displaystyle >}), and the inequality sign (≠ {\displaystyle \neq }). Unlike other expressions, statements can be true or false and their truth value usually depends on the values of the variables. For example, the statement x 2 = 4 {\displaystyle x^{2}=4} is true if x {\displaystyle x} is either 2 or − 2 and false otherwise. Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, like the equation 2 x + 5 x = 7 x {\displaystyle 2x+5x=7x}. Conditional equations are only true for some values. For example, the equation x + 4 = 9 {\displaystyle x+4=9} is only true if x {\displaystyle x} is 5.

The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as solving the equation for that variable. For example, the equation x − 7 = 4 {\displaystyle x-7=4} can be solved for x {\displaystyle x} by adding 7 to both sides, which isolates x {\displaystyle x} on the left side and results in the equation x = 11 {\displaystyle x=11}.

There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression 7 x − 3 x {\displaystyle 7x-3x} can be replaced with the expression 4 x {\displaystyle 4x} since 7 x − 3 x = (7 − 3) x = 4 x {\displaystyle 7x-3x=(7-3)x=4x} by the distributive property. Factorization is used to rewrite an expression as a product of several factors. This technique is commonly used to determine the values of a polynomial that evaluate to zero. For example, the polynomial x 2 − 3 x − 10 {\displaystyle x^{2}-3x-10} can be factorized as (x + 2) (x − 5) {\displaystyle (x+2)(x-5)}. The polynomial as a whole is zero if and only if one of its factors is zero, i.e., if x {\displaystyle x} is either − 2 or 5. For statements with several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that y = 3 x {\displaystyle y=3x} then one can simplify the expression 7 x y {\displaystyle 7xy} to arrive at 21 x 2 {\displaystyle 21x^{2}}. In a similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables.

Algebraic equations can be interpreted geometrically to describe spatial figures in the form of a graph. To do so, the different variables in the equation are understood as coordinates and the values that solve the equation are interpreted as points of a graph. For example, if x {\displaystyle x} is set to zero in the equation y = 0.5 x − 1 {\displaystyle y=0.5x-1}, then y {\displaystyle y} must be −1 for the equation to be true. This means that the (x, y) {\displaystyle (x,y)} -pair (0, − 1) {\displaystyle (0,-1)} is part of the graph of the equation. The (x, y) {\displaystyle (x,y)} -pair (0, 7) {\displaystyle (0,7)}, by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of (x, y) {\displaystyle (x,y)} -pairs that solve the equation.
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