Problem 67 - Algebra 1

Question

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

Step-by-Step Solution

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 1: Isolate Variable in the Second Equation

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

Step 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 \(0=0\)

Step 3: 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.

Step 4: 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.

Key Concepts

Infinitely Many Solutions

When we solve a system of linear equations and discover that the equation reduces to something like \(0 = 0\), it indicates that there are infinitely many solutions. This happens because the two original equations are essentially the same line in a different form. This means every point on the line is a solution to the system.

Here's what this implies for our exercise:

• The first equation and the second one describe the same line.
• There is no unique point of intersection because they overlap entirely.
• As a result, there's not just one solution or no solution – instead, there are infinitely many.

This concept is important because it shows us that linear equations can sometimes lead to not just one answer, but a whole set of answers that lie on the same line.

Dependent Equations

Dependent equations occur when two or more equations in a system represent the same line. This means that every solution of one equation is also a solution of the other. In the given exercise, when both equations simplify to the form \(0 = 0\), it confirms the system is dependent.

• The term "dependent" suggests reliance of one equation on the other for its solutions.
• Dependent equations lead to infinitely many solutions because solving them doesn't confine us to find a single point or none at all.
• This dependency indicates that the equations are linked, essentially conveying the same relationship between the variables.

Understanding dependent equations helps recognize the overlap in meaning between different equations in a system.

Parametric Form of Solution

When a system has infinitely many solutions, we can express these solutions using parameters. This is known as the parametric form of solution. By introducing a parameter, usually denoted as \(z\), we describe all solutions of the system in a compact way.

For the given system:

• From the second equation, by expressing \(y = x + 2\) and substituting in the dependency relation, one can rewrite it as \(x = z - 2\) and \(y = z\).
• Here, \(z\) represents any real number and is a free variable.
• This means you can pick any real number for \(z\), and the pair \((x, y)\) given by \(x = z - 2\) and \(y = z\) would be a solution to the system.

Parametric representations are powerful because they simplify a problem with infinitely many solutions into one that can be easily visualized and understood.