Question

Solve the system of linear equations if possible. Does the system have exactly one solution, no solution, or infinitely many solutions? $$ \begin{aligned}&-2 x+2 y=4\\\&x-y=-2\end{aligned} $$

Short answer

The given system of equations has infinitely many solutions, which can be parameterized as \(x=z-2\) and \(y=z\), where \(z\) is any real number.

Step-by-step solution

Isolate Variable in the Second Equation

From the second linear equation, isolate the variable \(y\) to get \(y = x + 2\)

Substitute into the First Equation

Substituting \(y = x + 2\) into the first equation, we get \(-2x + 2(x + 2) = 4\), which simplifies to \(-2x + 2x + 4 = 4\), and subtracting 4 from both sides gives us \(0=0\)

Analyse the Obtained Equation

The equation \(0=0\) is always true and tells us that the system of equations actually only contains one unique linear equation presented in multiple forms. This means that the equations are dependent, and there are infinitely many solutions.

Get the Parametric Form of Solution

Even though there are infinitely many solutions, we can express them in a parametric form. We substitute the obtained value of y into the second equation, yielding \(x=y-2\), or equivalently, \(x=z-2\) and \(y=z\), where \(z\) is an arbitrary parameter. This represents all possible solutions of the system.