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Understanding how to solve linear equations is fundamental in algebra, and it serves as a cornerstone for more advanced concepts in mathematics. A linear equation is an equation that makes a straight line when it is graphed. These equations typically look something like ax + by = c, where a, b, and c are constants.

When solving a linear equation, our goal is to find the value of the unknown variables that make the equation true. One of the ways to do this is by using an ordered pair, which is a pair of numbers that can represent a point in a coordinate plane. An ordered pair is written in the form (x, y). To determine if a specific ordered pair is a solution to an equation, we substitute the values of x and y into the equation and simplify. If the equation holds true after the substitution, then the ordered pair is indeed a solution.

For example, to check if the ordered pair (-2, 1) is a solution, substitute -2 for x and 1 for y into the equation -3x - 8y = 14. Simplifying the expression by following the order of operations will confirm if the equality stands, thus verifying the solution.

The substitution method is a useful technique in algebra, especially when it comes to solving systems of equations or verifying solutions. It involves replacing a variable with a given value or another expression. While solving multi-variable equations, if one variable's value is known, it can be substituted in the place of that variable in the equation to find the value of the other variable.

To apply this method properly, it is important to perform the substitution accurately and ensure that the algebraic operations that follow are carried out correctly. In the problem at hand, we are substituting -2 for x and 1 for y in the equation -3x - 8y = 14. After the substitution, we perform algebraic operations to simplify the expression and check if the result is equal to the constant on the other side of the equation.

• If after simplification, both sides of the equation are equal, the substitution is verified,
• If not, the ordered pair is not a solution.

Algebraic operations include the fundamental processes of addition, subtraction, multiplication, and division, applied to algebraic expressions. When verifying a solution to an equation, these operations are used to simplify the expression after substituting the values of the variables.

It's crucial to follow the proper order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right)). In the given task, after substituting the values into the equation -3x - 8y = 14, we simplify to find 6 - 8. By subtracting 8 from 6, we get -2, which is not equal to 14, thus indicating the ordered pair (-2, 1) is not a solution to the given linear equation. This demonstrates the importance of performing algebraic operations correctly to verify the solutions accurately.