Algebraic Operations and Linear Equations in Mathematics

Mathematics - Algebra

Mathematics is a subject that involves various branches, one of which is algebra. This branch deals with symbols and the rules for manipulating these symbols. Here's what you need to know about algebra within mathematics:

Basic Operations in Algebra

Algebra is not just about letters; it also includes numbers. It uses symbols like 'x', 'y', and 'z' instead of numbers. These symbols represent values. In algebra, we do basic operations like addition, subtraction, multiplication, division, exponentiation, logarithms, etc., which involve these symbols.

Here's how you can perform basic operations in algebra:

Adding Algebraic Expressions

To add two algebraic expressions, you simply combine their coefficients if they have the same variable raised to a power. If the variables have different powers, the expression cannot be added.

Example

x + y = x + y

Subtracting Two Algebraic Expressions

Subtracting two algebraic expressions works similarly to adding them, except that you switch the signs of the coefficients.

Example

x - y = y - x

Multiplying Two Algebraic Expressions

Multiplying two algebraic expressions means multiplying each term of one equation by each term of another equation.

Example

(x + y)(a + b) = ax + ay + by + bx

Dividing Two Algebraic Expressions

Dividing one algebraic expression by another means dividing each term by corresponding terms.

Example

(x + y)/(a + b) = x/(a + b) + y/(a + b)

Exponentiating an Algebraic Expression

Exponentiating an algebraic expression means raising each term to the appropriate power.

Example

(x + y)^2 = x^2 + 2xy + y^2

Logarithmizing an Algebraic Expression

Logarithmizing an algebraic expression means taking the logarithmic value of each term.

Example

log_(a+b)(x + y) = log_ax + log_by

Solving Linear Equations

Solving linear equations in algebra involves finding all the possible values of the unknowns for a given set of equations. There are several methods to solve these equations, such as substitution, elimination, cross-multiplication, or the matrix method.

For example, consider the system of equations below:

2x + 3y = 10 x - y = 1

To find the solutions, we can use any of the above mentioned methods. Let's use the substitution method:

From the second equation, we get y = x - 1. Now, substitute this into the first equation:

2x + 3(x - 1) = 10 => 2x + 3x - 3 = 10 => 5x = 13 => x = 13/5

Now, plug x back into the second equation:

13/5 - (x - 1) = 1 => 13/5 - x + 1 = 1 => 14/5 - x = 1 => x = 14/5 - 1/5 = 13/5

The solution is x = 13/5 and y = x - 1 = 12/5 - 1/5 = 11/5.

Therefore, the solution of this system is (13/5, 11/5).