To solve the system of equations:
[tex]\[
\begin{cases}
x + y = 3 \\
-2x - 2y = -6
\end{cases}
\][/tex]
we can follow these steps:
1. Simplify the Second Equation:
Let's first simplify the second equation by dividing everything by -2:
[tex]\[
-2x - 2y = -6 \implies x + y = 3
\][/tex]
Notice that after simplification, the second equation is the exact same as the first equation.
2. Recognize Dependence:
Since both equations [tex]\( x + y = 3 \)[/tex] essentially represent the same line, this means we are dealing with dependent equations. In other words, one equation is a multiple of the other, indicating that they represent the same line in a coordinate plane.
3. Solve for One Variable in Terms of the Other:
From either of the simplified equations [tex]\( x + y = 3 \)[/tex], we can express one variable in terms of the other. Let's solve for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[
x = 3 - y
\][/tex]
The solution to the system of equations is therefore:
[tex]\[
x = 3 - y
\][/tex]
Hence, the variable [tex]\( x \)[/tex] is dependent on [tex]\( y \)[/tex] and is represented as [tex]\( x = 3 - y \)[/tex]. For any value of [tex]\( y \)[/tex] that satisfies the equation, there will be a corresponding [tex]\( x \)[/tex] value given by this relationship.
