91Ó°ÊÓ

A system of equations consists of multiple equations that have common variables, which you need to solve simultaneously. The objective is to find values for these variables that satisfy all the equations in the system at the same time. Systems of equations can appear in two variables, usually denoted as \( x \) and \( y \), or even more in complex cases.
In our given exercise, the system consists of two equations:

• \( x + y = -5 \)
• \( 2x - y = 2 \)

The solution to this system is the pair \( (x, y) \) that makes both equations true at once. Understanding this concept is crucial for solving various mathematical problems that involve finding intersections, modeling real-world situations, and more. Solving systems can be done by different methods such as graphing, substitution, and elimination.

The substitution method is one of the valuable techniques in solving systems of equations. It involves expressing one variable in terms of the other using one equation and then substituting this expression into the other equation. This method effectively reduces the system to a single equation in one variable, making it easier to solve.
In our example, we started with the equation \( x + y = -5 \) and rearranged it to get an expression for \( y \):

• \( y = -5 - x \)

This expression was then substituted into the second equation \( 2x - y = 2 \), transforming it into a single equation with only \( x \).
The key advantage of the substitution method is its simplicity and directness, particularly when one equation is already easily solvable for one variable. Through practice, you'll become adept at deciding which equation to rearrange and substitute, a skill that simplifies complex algebraic problems.

Problem-solving in mathematics goes beyond just finding the right answer; it's about understanding and applying the correct methods to approach a problem. Solving systems of equations by substitution is a classic exercise in applying logical reasoning and systematic thinking.
The process is often broken down into clear, manageable steps:

• Rearrange one of the equations to solve for a variable.
• Substitute this into the other equation.
• Solve the resulting single-variable equation.
• Use the solution to find the value of another variable.
  
  After solving, verification is crucial. Plug your values back into the original equations to ensure they hold true. Not only does this confirm the correctness, but it also reinforces your understanding of the problem.
  Practicing problem solving helps build confidence and improves the ability to tackle more complex mathematics scenarios, nurturing critical thinking skills along the way.