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An ordered triple is a set of three numbers used to represent the coordinates of a point in three-dimensional space, typically written as \(x, y, z\). In the context of systems of equations, these triples can serve as potential solutions. Each number corresponds to a variable in the system. For example, if we have an ordered triple (a, b, c), then we substitute \(x = a\), \(y = b\), and \(z = c\) into our equations to see if they satisfy all of them. This process allows us to determine whether a specific point is a solution to all the equations in a three-variable system.
For instance, taking the ordered triple (1, 2, 6):
\ \begin{align*} -x - y + z &= 3 & \rightarrow -1 - 2 + 6 = 3 \ 3x + 4y - z &= 1 & \rightarrow 3(1) + 4(2) - 6 = 1 \ 5x + 7y - z &= -1 & \rightarrow 5(1) + 7(2) - 6 = -1 \ \end{align*} As you can see, substituting (1, 2, 6) into each equation makes them true. So, (1, 2, 6) is indeed a solution. Using ordered triples is a structured way to test potential solutions in three-dimensional systems.

The substitution method is a technique for solving systems of equations where one of the equations is solved for one variable in terms of the others, and then substituted into the remaining equations. This helps reduce the number of variables in the remaining equations and makes the system easier to solve step by step.
Let's break it down for a system of three equations:

• Step 1: Solve one of the equations for one variable.
• Step 2: Substitute that expression into the other equations.
• Step 3: Solve the resulting system of simpler equations.
• Step 4: Use the found values to solve for the last variable.

For example, consider three equations:
\ \begin{align*} -x - y + z &= 3 \ 3x + 4y - z &= 1 \ 5x + 7y - z &= -1 \ \end{align*} Suppose we solve the first equation for \(z\):
\[ z = x + y + 3 \] We then substitute \(z = x + y + 3\) into the second and third equations to eliminate \(z\). This will give us two new equations in terms of \(x\) and \(y\). Solving those will give us values for \(x\) and \(y\), and finally for \(z\). The substitution method is handy because it simplifies complex systems into more manageable parts.

Linear algebra is a branch of mathematics dealing with vectors, vector spaces, and linear equations. It provides essential tools for understanding systems of equations, including those involving ordered triples. Systems of linear equations can be represented with matrix equations, such as \(AX = B\), where \(A\) is a matrix of coefficients, \(X\) is a column matrix of variables, and \(B\) is a column matrix of constants.

For example, consider a system:
\ \begin{align*} -x - y + z &= 3 \ 3x + 4y - z &= 1 \ 5x + 7y - z &= -1 \ \end{align*} We can write this in matrix form:
\[ \begin{pmatrix} -1 & -1 & 1 \ 3 & 4 & -1 \ 5 & 7 & -1 \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 3 \ 1 \ -1 \end{pmatrix} \] Using various techniques from linear algebra, such as Gaussian elimination or matrix inversion, we can solve for the variables to find the ordered triple solutions. These techniques often streamline solving large systems of equations that might otherwise be very complex. Understanding linear algebra fundamentals is crucial for higher-level mathematics and various applications in physics, engineering, and computer science.