91Ó°ÊÓ

A system of equations comprises two or more equations with a shared set of variables. An essential part of algebra, these systems are used to find the values of these variables that satisfy all equations simultaneously. In the given exercise, we have two equations involving \(x\) and \(y\):

• \(6x + y = -6\)
• \(-12x - 2y = 12\)

The main goal is to find values of \(x\) and \(y\) that make both equations true at the same time.

There are various methods to solve a system of equations, such as substitution, elimination, and graphing. Here, we focus on the substitution method because it involves solving one equation for one variable and then substituting that expression into the other equation. This method is particularly effective in situations where one of the variables has a coefficient of one, as it simplifies the algebraic manipulation.

Infinite solutions occur in a system of equations when you end up with a true statement that doesn't involve any variables, like \(0 = 0\) or \(12 = 12\). Essentially, it means that both equations represent the same line, so they overlap completely. In other words, every point on the line is a solution.

In our exercise, after substitution and simplification, we had:

• \(-12x + 12 + 12x = 12\)

Simplifying leads to \(12 = 12\), indicating that any value of \(x\) yields a valid solution since it leads to an identical equation. This is how we know the system has infinite solutions.

Understanding infinite solutions is crucial, as it brings awareness to situations where finding just one variable's value won't suffice. Instead, it implies a relationship, or a functional dependency between the variables. In this case, \(y = -6 - 6x\), which is the equation of the line where all solutions lie.

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation signs. In the substitution method, creating an expression from one equation to replace a variable in another is key. Consider the expression derived in the problem’s step 1:

\(y = -6 - 6x\)

This expression for \(y\) was obtained by rearranging the first equation, simplifying our work by allowing us to substitute it straight into the second equation.

Algebraic expressions like this one are powerful, as they maintain the relationships between variables and simplify the process of solving complex systems. Being able to manipulate these expressions makes them an excellent tool in the algebra toolbox. It not only aids in solving systems but also helps in understanding the inherent relationships present in equations.