91Ó°ÊÓ

Algebra is an essential branch of mathematics that deals with symbols and the rules for manipulating those symbols. It forms the basis for more complicated mathematical concepts and is used extensively in various fields, such as engineering, physics, and economics. In algebra, we often use letters to represent variables or unknown values, and then work with those variables through equations. An equation is a mathematical statement that asserts that two expressions are equal. The objective of solving an algebraic equation is to find the value of the variable that makes the equation true.

Sometimes, algebraic equations involve multiple variables and terms. In such cases, we use operations such as addition, subtraction, multiplication, and division to simplify or manipulate the equation and isolate the variables.

Isolating variables means rearranging an equation so that a particular variable is alone on one side of the equation. This makes it easier to solve for the variable and to understand how changes in other variables affect it.

The basic steps to isolate a variable include:

• Identifying the term that contains the variable.
• Using inverse operations to move other terms to the opposite side of the equation. Inverse operations are operations that undo each other, like addition and subtraction or multiplication and division.
• Simplifying the equation to isolate the variable completely.

For example, in the given exercise, we need to isolate x in the equation:
\( A x + B y = C \)
First, subtract By from both sides to get:
\( A x = C - B y \)
Then, divide by A to isolate x:
\( x = \frac{C - B y}{A} \)

Linear equations are equations of the first degree, which means they involve only the first powers of variables and result in straight lines when graphed. They typically have the form \(ax + by = c\), where a, b, and c are constants, and x and y are variables.

Linear equations can have one variable, two variables, or more, but the solutions are always straight lines in a coordinate system.
In the given exercise, we worked with a linear equation in two variables:
\(A x + B y = C\)
This equation represents a straight line in the xy-plane. To solve for a specific variable like x, we need to isolate it by following the steps mentioned. This involves subtracting \(By\) from both sides and then dividing both sides by A to get:
\( x = \frac{C - B y}{A} \)
By isolating x, we can see how x changes as y changes, which is a fundamental aspect of understanding linear relationships and functions.

Linear equations are foundational in algebra and serve as stepping stones for studying more complex mathematical concepts.