## Introduction to Linear Equations

Linear equations form the basis of many algebraic applications, from calculating distances and investments to resolving supply and demand in economics. A linear equation in two variables is any equation that can be written in the form:

where , , are the coefficients and is a constant, and and are variables. When dealing with pairs of linear equations, we are interested in finding the values of and that satisfy both equations simultaneously.

For example, consider the system of linear equations:

Our goal is to find values of and that make both equations true.

## Methods of Solving Linear Equations

There are several algebraic methods for solving pairs of linear equations, including:

- Graphical Method
- Substitution Method
- Elimination Method
- Matrix Method

While graphical methods provide a visual understanding, algebraic methods such as substitution and elimination are more precise and commonly used.

## Graphical Method

The graphical method involves plotting both equations on a coordinate system and finding the intersection point. The point of intersection is the solution to the system of equations.

For example, let’s revisit our previous system:

Solving each equation for , we get:

Thus, the solution is and .

The intersections of these lines on a graph provide the solution. While not always precise for complex systems, it’s a good method for visual learners.

## Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting that expression into the other equation. This reduces the system to a single equation with one variable.

For example, in the system:

Solving the second equation for :

Substitute this value into the first equation:

This simplifies to:

Substitute back into :

Thus, the solution is and .

## Elimination Method

The elimination method involves adding or subtracting equations to eliminate one variable, making it simpler to solve for the remaining variable.

Consider the system:

We can multiply the second equation by 3 to align the coefficients of :

Adding these equations:

Once we have , substitute it back into one of the original equations:

Thus, the solution is and .

## Real-world Application: Economics

Consider a real-world scenario where you run a business selling two products, and . Suppose the revenue equation for each product is:

Here, represents the number of product , and represents the number of product . Solving these simultaneous equations can help determine how many units of each product should be sold to maximize profits while covering costs. By substituting or eliminating variables as shown, you can find the optimal quantities of and .

## Conclusion

Understanding and applying algebraic methods to solve pairs of linear equations is a fundamental skill in mathematics. These techniques, such as substitution and elimination, empower students to approach real-world problems methodically. By mastering these methods, you can tackle more complex scenarios, from business economics to engineering solutions, reinforcing the practical value of mathematics in everyday life.