91Ó°ÊÓ

An ordered pair is simply a set of two numbers written in a specific order, usually as \( (x, y) \). The first number represents the x-coordinate, and the second number represents the y-coordinate. Ordered pairs are used to locate points on a coordinate plane. For example, in the exercise given, the pairs \( (1, 7/6) \) and \( (2, 3) \) are ordered pairs. They indicate specific points that can be checked to see if they are solutions to a given system of equations. Checking whether an ordered pair is a solution involves substituting these values into each equation in the system to verify if the equations are satisfied.

The substitution method is one way of solving a system of equations. The idea is to solve one of the equations for one variable and then substitute this in the other equation. Let's go through the exercise to see how the substitution method works.

In our original exercise, we're given the system of equations:
\(-11x + 6y = -4\) and \(7x + 3y = 23\)

To determine if the ordered pair \( (1, 7/6) \) is a solution, we substitute \(x = 1\) and \(y = 7/6\) into both equations.

First, substituting into the first equation: \(-11(1) + 6(7/6) = -11 + 7 = -4\) (True).
Second, substituting into the second equation: \(7(1) + 3(7/6) = 7 + 7/2 = 21/2\) (False).

Since the ordered pair satisfies one equation but not the other, it's not a solution.

The same process is followed for the pair \( (2, 3) \), which turns out to be a valid solution for both equations. Hence, we see how substituting the ordered pair into each equation helps verify if it satisfies the system.

A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The general form of a linear equation in two variables is \(Ax + By = C\), where A, B, and C are constants.

In a system of linear equations, we have two or more such equations, and we are looking for solutions that satisfy all equations simultaneously.

In our exercise, the system given is:
\[-11x + 6y = -4\]
\[7x + 3y = 23\]

We solve this system by checking potential solutions and verifying if they satisfy both linear equations. This can be done graphically by finding the point where the lines intersect or algebraically using methods such as substitution or elimination.

In our specific example, solving the system involves substituting the values of x and y from each ordered pair and simplifying to see if the original equation holds true. If both equations are satisfied with the same ordered pair of \(x, y\), then we say the pair is a solution to the system of linear equations.