91Ó°ÊÓ

A system of equations is essentially a set of two or more equations that have the same set of unknowns. In our given problem, the system is made up of two linear equations:

• \( x - y = 2 \)
• \( 2x + 3y = 9 \)

The goal of solving a system of equations is to find the values of the variables that satisfy each equation at the same time.
The variables here are \( x \) and \( y \), and the solution is the point where these lines intersect on a graph, representing a common solution to both equations.
When dealing with systems of equations, there are several methods you can use to solve them:

• Substitution method
• Elimination method
• Graphical method

In this particular exercise, we will use the substitution method to find the solution.

To solve for \( x \) using the substitution method, you first isolate \( x \) in one of the equations. In our system, we use the first equation:

• Start with: \( x - y = 2 \)
• Rearrange the equation to solve for \( x \): \( x = y + 2 \)

This gives us an expression where \( x \) is expressed in terms of \( y \).
We then use this expression to substitute into the other equation. This substitution allows us to eliminate \( x \) and focus on solving for \( y \). It’s a strategic move that simplifies the system gradually by reducing the number of variables involved, making it easier to find a solution.

After substituting \( x = y + 2 \) into the second equation \( 2x + 3y = 9 \), we obtain:
\[2(y + 2) + 3y = 9\]
This results in a new equation with only \( y \) as a variable:

• Expand the equation: \( 2y + 4 + 3y = 9 \)
• Combine like terms: \( 5y + 4 = 9 \)
• Solve for \( y \) by subtracting 4 from both sides: \( 5y = 5 \)
• Finally, divide both sides by 5: \( y = 1 \)

Now, we have found the value of \( y \). This step is crucial as it provides one part of the solution needed to solve the entire system.

Once you've determined the value of \( y \), the next step is to find \( x \) to complete the ordered pair solution. A solution to the system of equations is typically expressed as an ordered pair \((x, y)\), representing the values of \( x \) and \( y \) that satisfy both equations simultaneously.

Substituting \( y = 1 \) back into the expression for \( x \) from our earlier step, \( x = y + 2 \), we substitute:

• \( x = 1 + 2 \)
• This gives: \( x = 3 \)

Combining these values provides us with the ordered pair solution:

• \((x, y) = (3, 1)\)

The ordered pair \((3, 1)\) tells us that when \( x = 3 \) and \( y = 1 \), both equations in the system are satisfied. This ordered pair represents the intersection point of the two lines denoted by the equations when graphed.