91Ó°ÊÓ

A system of equations consists of two or more equations with the same set of variables. The goal of solving a system of equations is to find the values of these variables that satisfy all equations simultaneously. In the given exercise, we have a system of two linear equations in two variables, \(x\) and \(y\). Each equation represents a line on a coordinate plane.

There are three possible outcomes when solving a system of equations:

• *Unique Solution*: The lines intersect at one point, indicating a single solution for \(x\) and \(y\).
• *Infinitely Many Solutions*: The lines coincide, meaning they are the same line, resulting in infinitely many solutions.
• *No Solution*: The lines are parallel and never intersect, indicating no common points of intersection and no solution.

In our exercise, we determined no solution exists because the equations represent parallel lines. This was deduced by examining the ratios of the coefficients.

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear equations. It provides powerful methods to solve systems of linear equations, which is essential in various fields like engineering, physics, computer science, and economics.

Consider the system from the exercise, represented by two linear equations. We can express each equation in the form \(ax + by = c\), comprising three components:

• **Coefficients**: The numbers multiplying the variables \(x\) and \(y\).
• **Variables**: The unknowns \(x\) and \(y\) we are solving for.
• **Constant terms**: The numbers on the right side of the equations.

By analyzing the coefficients and constants, primarily through matrix representation and manipulation, linear algebra helps to determine whether the system is consistent (having solutions) or inconsistent (having no solutions). In this context, our system lacked a solution because we found that the equations are parallel lines.

In geometry, parallel lines are defined as two lines in the same plane that never intersect, no matter how far they are extended. This geometric property plays a crucial role when interpreting the nature of the solutions of a system of equations.

When two linear equations form parallel lines, it indicates a lack of intersection points, meaning there is no common solution to the equations. In practical terms, we determine this parallelism by comparing the slopes of the two lines.

To find the slope of a line given by an equation in the form of \(ax + by = c\), rearrange it to \(y = mx + b\), where \(m\) represents the slope. If two lines have the same slope, they are parallel.

In our exercise, we established that the lines are parallel by checking the ratios of the coefficients for \(x\) and \(y\). Although these were equal, the differing constant terms confirmed that the lines never meet, thus confirming their parallel nature and leading to the conclusion of no solution for the system.