Question

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} x+2 y=4 \\ 2 x+4 y=8 \end{array}\right. $$

Short answer

Infinitely many solutions; the system is consistent and the equations are dependent.

Step-by-step solution

Identify the System of Equations

The given system of equations is: \ 1. \(x + 2y = 4\) \ 2. \(2x + 4y = 8\)

Rewrite Second Equation

Observe that the second equation is a multiple of the first equation. Divide the second equation by 2 to compare it directly with the first equation: \ \(2x + 4y = 8\) becomes \(x + 2y = 4\)

Compare the Two Equations

Notice that both equations now look identical: \ First: \(x + 2y = 4\) \ Transformed Second: \(x + 2y = 4\)

Analyze the Equations

Since both equations are identical, any solution that satisfies one equation will satisfy the other. Therefore, there are infinitely many solutions to this system. It's a consistent system with dependent equations.

Key concepts

consistent system

A system of equations consists of two or more equations that share common variables. When we say a system is 'consistent,' it means that there is at least one set of values for the variables that satisfies all equations in the system. In other words, the equations do not contradict each other.

In our example, the system of equations given is:
\(x + 2y = 4\)
\(2x + 4y = 8\)

By transforming the second equation, we can see that it is essentially the same as the first one. Therefore, any solution that works for one equation will also work for the other.

This means the system is consistent because it has infinitely many solutions. If there had been no possible solution that could satisfy both equations, we would call the system 'inconsistent.' However, that is not the case here.

dependent equations

Dependent equations are equations in a system that can be derived from each other. Essentially, one equation is a multiple of the other.

In our given system, observe that the second equation is exactly twice the first equation:
\(2x + 4y = 8\) is simply \(2(x + 2y) = 2 \times 4\)

After dividing the second equation by 2, we get:
\(x + 2y = 4\)

Now both equations are identical:
\(x + 2y = 4\) and \(x + 2y = 4\)

The identical nature of these equations means they are dependent. If you solve one, you have also solved the other. This dependency means the system will either have infinitely many solutions (if consistent) or no solution at all (if inconsistent).

solution of a system

Solving a system of equations means finding the values of the variables that satisfy all the equations simultaneously. With consistent systems, as in our example, identifying solutions can be straightforward.

First, we recognize the given system:
\(x + 2y = 4\)
\(2x + 4y = 8\)

We already know that these equations are dependent because they are multiples of each other. Thus, we look for solutions for just one of them.

Since \(x = 4 - 2y\), any value of \(y\) will provide a corresponding \(x\) that satisfies the equation. For instance:

• When \(y = 0\): \(x = 4\)
• When \(y = 1\): \(x = 2\)
• When \(y = -1\): \(x = 6\)

These values form pairs \((x, y)\) that serve as solutions for the system, demonstrating it has an infinite number of solutions.

Thus, the solution of a system can be a single unique set of values (for independent systems), multiple sets (for dependent and consistent systems), or none (for inconsistent systems).