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In the world of mathematics, linear equations are fundamental and play a crucial role in various applications. One of the standard forms of linear equations is ax + by = c, where ‘a,’ ‘b,’ and ‘c’ are constants, while ‘x’ and ‘y’ are variables. In this article, we will explore the intricacies of this linear equation, its applications, and methods to solve it.

## The Basics of Linear Equations

A linear equation is a fundamental algebraic expression where every term has an exponent of one, producing a straight line when plotted on a graph. It is aptly named due to its linear nature. Two common types of linear equations are linear equations in one variable and linear equations in two variables.

In our focus equation, ax + by = c, ‘a’ and ‘b’ are coefficients, ‘x’ and ‘y’ are variables, and ‘c’ is a constant. This equation represents a straight line on a graph.

## Key Terms and Concepts

A linear equation in one variable has a basic form of ax + c = 0, where ‘a’ is a coefficient, ‘x’ is the variable, and ‘c’ is a constant. Solving such equations involves isolating ‘x’ to find its value.

In the context of ax + by = c, ‘a’ and ‘b’ are coefficients, ‘x’ and ‘y’ are variables, and ‘c’ is a constant. Solving linear equations in two variables requires techniques such as substitution, graphical methods, cross multiplication, and elimination.

### Resolving Two Linear Equation Systems

One significant application of ax + by = c is its usefulness in solving systems of two linear equations. When you have two equations in this form, you can find the values of ‘x’ and ‘y’ that satisfy both equations simultaneously.

## Methods of Solving Linear Equations

*Solving a Linear Equation in One Variable*

Solving a linear equation in one variable, like ax + c = 0, is relatively straightforward. By isolating ‘x,’ you can find its value. For example, to solve the equation 2x + 3 = 7, you subtract 3 from both sides, resulting in 2x = 4, and then divide by 2 to get x = 2.

*Solving a Linear Equation in Two Variables*

Solving ax + by = c, where ‘a’ and ‘b’ are coefficients, ‘x’ and ‘y’ are variables, and ‘c’ is a constant, is more complex. Various methods can be employed, including graphical methods, substitution, cross multiplication, and elimination.

*Graphical Method*

Graphically representing linear equations involves plotting them on a graph. In the context of ax + by = c, if you have a system of two linear equations, you can find solutions by locating their intersection points on the graph. There are three types of solutions:

- Unique Solution: When two lines intersect at a single point, there’s one unique solution.
- Infinitely Many Solutions: If two lines coincide, there are infinitely many solutions along the line.
- No Solution: When two lines are parallel, there is no solution.

*Substitution Technique*

The substitution technique is an algebraic approach to solving systems of linear equations in two variables. It involves isolating one variable in one equation and substituting it into the other equation.

*Cross Multiplication Technique*

The cross multiplication technique simplifies solving linear equations with two variables. It is a straightforward method used in such equations.

*Elimination Technique*

By using basic arithmetic operations, the elimination technique allows you to eliminate one variable and simplify the equation to find the value of the second variable.

## Conclusion

In conclusion, the linear equation ax + by = c is a fundamental mathematical concept with wide-ranging applications. Whether you’re dealing with linear equations in one variable or two, understanding the basics and employing various solving methods such as graphical representation, substitution, cross multiplication, and elimination can help you find solutions efficiently. This knowledge is essential in various fields, including physics, engineering, and economics, where linear equations play a vital role in modeling and problem-solving.

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