To solve the system of equations:

$-2x = 2 - y$ $2y = 4x + 4$

we'll use substitution or elimination to find values for $x$ and $y$.

Step 1: Rearrange the Equations

Let's start by rearranging the first equation to express $y$ in terms of $x$.

1. From the first equation: $-2x = 2 - y$ Add $y$ to both sides: $y - 2x = 2$ This can be written as: $y = 2x + 2$

Now substitute this expression for $y$ into the second equation.

Step 2: Substitute and Solve for $x$

2. Substitute $y = 2x + 2$ into the second equation: $2(2x + 2) = 4x + 4$ Expanding: $4x + 4 = 4x + 4$

This equation holds true for all $x$ and doesn't isolate a particular value of $x$. This implies the system of equations is dependent, meaning the equations represent the same line and therefore have infinitely many solutions.

Solution

Since both equations represent the same line, any pair $(x, y)$ that satisfies $y = 2x + 2$ is a solution to the system.

Would you like more details or have any questions?

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Here are some questions to explore:

1. How do we determine if a system of equations has one solution, no solution, or infinitely many solutions?
2. What are the differences between independent, dependent, and inconsistent systems?
3. How can we use graphing to understand solutions to a system of linear equations?
4. What does it mean for two equations to be equivalent in a system of equations?
5. How would this system change if we had different coefficients in either equation?

Tip: When both equations simplify to the same line, the system is called "dependent" and has infinitely many solutions.