91Ó°ÊÓ

Let's start by understanding **linear equations**. A linear equation is an algebraic equation of the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. The graph of a linear equation forms a straight line. In our exercise, the original equations are linear equations: \( \frac{1}{2} x - \frac{1}{6} y = 10 \) and \( \frac{2}{5} x + \frac{1}{2} y = 8 \).

The coefficients of \( x \) and \( y \) ((\frac{1}{2}, \frac{1}{6}) and (\frac{2}{5}, \frac{1}{2})) determine the steepness and direction of the lines. By eliminating fractions and solving for variables, these types of equations can be simplified and solved methodically.

A **system of equations** consists of two or more equations with the same set of variables. In our case, we have two equations and two variables, \( x \) and \( y \). The goal in solving a system of equations is to find the values of the variables that satisfy all equations simultaneously. There are three possible outcomes when solving a system:

• One unique solution
• No solution
• An infinite number of solutions

Our example uses the elimination method to find the unique solution to the system. By manipulating the equations, we can align the coefficients of one of the variables and then eliminate it. This allows us to solve for the remaining variable.

Once one variable is found, we substitute it back into one of the equations to find the other variable.

When describing the solution set of a system of equations, especially when it has infinite solutions, we use **set-builder notation**. This notation allows us to specify a set of numbers that satisfy a certain condition. It is expressed as:

\( \{ x \, | \, \text{condition} \} \)

For example, if the solution set of a system is all pairs \( (x, y) \) that satisfy a particular linear equation, we would write something like:

\( \{ (x, y) \, | \, y = 2x + 3 \} \).

In our exercise, we found a unique solution. Hence, we didn't need set-builder notation. But, if the system had infinite solutions, we would use this notation to describe all possible solutions.

Finally, let's break down the steps involved in **solving algebraic equations** using the elimination method. This process helps isolate variables and find solutions:

• First, eliminate fractions by finding common denominators and multiplying through to simplify the equations.


• Next, align the coefficients of one of the variables by multiplying the entire equation by appropriate constants.


• Subtract or add the equations to eliminate one variable, resulting in a single-variable equation.


• Solve this equation to find the value of the remaining variable.


• Substitute this value back into one of the original equations to solve for the other variable.

By following these steps, we ensure that we find a valid solution to the system of equations. In the given exercise, we used these steps to find that \( x = 20 \) and \( y = 0 \), indicating a unique solution to the system.