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A system of equations is essentially a set of two or more equations that share two or more variables. It is used to find the values of these variables that make all equations true at the same time.
In our example, the system of linear equations derived from the matrix can be written as:

• Equation 1: \(x - 4u = -3\)
• Equation 2: \(y - 2u = 1\)
• Equation 3: \(z + 3u = -10\)

These equations stem from the rows of the matrix where each row directly translates into a linear equation, using the order of variables given. Remember in a matrix, each row correlates to one equation, and the column under each variable represents its coefficient in the equation.
Answering systems of equations involves determining the values of variables that satisfy all listed equations. This can often be done using methods like substitution, elimination, or matrix row transformations depending on the form the system is in.

The consistency of a system refers to whether a system of equations has solutions or not. A system can either be:

• Consistent: Has at least one solution. This means that all equations align well enough to have a common solution point.
• Inconsistent: Has no solutions. This occurs if, for example, the equations represent parallel lines that never intersect.

To determine consistency in the example, we look for the presence or absence of a contradictory equation like \(0 = a\) (where \(a\) is non-zero) after transforming the matrix to row-echelon form. In this specific instance, the absence of such a row implies that the system is consistent.
Because we have variables that outnumber equations after simplifying, the system doesn’t pinpoint a single solution, indicating infinite solutions constrained by the relationships across equations.

Finding the solution to a system of linear equations involves identifying values for each variable that satisfy every equation in the system. For systems with more variables than equations, like this case, expect infinitely many solutions expressed in terms of a free variable.
Here, solving for \(x\), \(y\), and \(z\) involves their expressions in terms of \(u\), which is a free variable.

• From \(x - 4u = -3\), we derive \(x = 4u - 3\).
• From \(y - 2u = 1\), we derive \(y = 2u + 1\).
• From \(z + 3u = -10\), we derive \(z = -3u - 10\).

Each of these solutions depends on the chosen value of \(u\), highlighting that for every value of \(u\), there are specific, corresponding values for \(x\), \(y\), and \(z\) that satisfy the system. This concept of expressing solutions in parametric form illustrates the nature of dependent solutions within linear equation systems.