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Understanding the concept of ordered pairs is crucial when dealing with systems of equations. An ordered pair, usually written as \(x, y\), is a pair of numbers that represent the coordinates of a point on a two-dimensional plane. The first number, \(x\), refers to the horizontal position, while the second number, \(y\), indicates the vertical position.

For example, in our exercise, the ordered pair \(2, -3\) corresponds to a point where \(x=2\) and \(y=-3\). To determine if this ordered pair is a solution to the given system of equations, we test this pair in each equation. If the pair satisfies both equations, then it represents a point where the lines represented by the equations intersect, identifying it as a solution to the system.

Algebraic substitution is a method used to solve systems of equations. It involves replacing variables with their corresponding values to simplify equations and solve for unknowns. In our problem, we substitute the values from the ordered pair directly into the equations.

For instance, substituting \(x = 2\) and \(y = -3\) into the equations gives us two expressions: \(2(2) + 3(-3)\) for the first equation and \(7(2) - 3(-3)\) for the second. When we perform the calculations, if both simplified expressions match the constants on the right side of the equations (in this case, -5 and 23, respectively), we confirm that \(2, -3\) is indeed a solution to the system. Algebraic substitution is a straightforward technique that can be used effectively when dealing with linear systems.

Solving linear equations lies at the heart of finding solutions to systems of equations. A linear equation represents a straight line and can be solved for one variable in terms of another. The general form is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.

When we plug an ordered pair into a linear equation, we are essentially verifying whether the corresponding point lies on the line that the equation represents. If the left side of the equation equals the right side after substituting the values, then the equation is true for that pair, indicating that the pair is on the line. To solve the equation completely, one would typically isolate one variable and solve the equation in terms of the other variable, but with an ordered pair given, we only need to check if the substitution holds true.

Simultaneous equations are a set of equations that are solved together, as they share common variables. The solution to a system of simultaneous equations is the ordered pair that satisfies all equations in the system at once. There are various methods to solve these systems, such as graphing, substitution, elimination, and matrix operations.

In our exercise, we deal with a pair of simultaneous linear equations. By substituting the given ordered pair into each equation, we test for a common solution. If the ordered pair does not satisfy even one of the equations, it cannot be the solution to the system. The concept is similar to finding the intersection point of two lines; if they intersect, they share a common point, which is the solution to the system of equations.