91Ó°ÊÓ

Ordered pairs are a simple and powerful way to represent values on a coordinate system. They consist of two elements, usually written as \(x, y\). The first element represents the position on the x-axis, and the second element represents the position on the y-axis. In the context of linear equations, ordered pairs can show us how the x and y values relate to each other through the equation.
For example, when using the equation \(y = x + 5\), we can substitute different values for \x\ to find the corresponding \y\ values. Each pair of \(x, y\) found is an ordered pair that is a solution to the equation. In our exercise, each value we calculate for \y\ corresponds to the x-values \{-3, -1, 0, 1\}, creating these ordered pairs: \(-3, 2\), \(-1, 4\), \(0, 5\), and \(1, 6\). These points can then be plotted on a graph to show a linear relationship which forms a straight line when connected.
This structured way of using pairs helps in visualizing and understanding the relationship between variables in a system.

Table completion is a useful method for organizing and calculating values, especially when dealing with linear equations. By setting up a table, you can systematically find solutions for different values of a variable and see their effects on an equation's output.
Our exercise table lists different x-values: \{-3, -1, 0, 1\}. The column labeled \(x+5\) helps guide in calculating \y\ based on the linear equation \(y = x + 5\).
Tables in mathematics serve various purposes:

• Organizing data clearly for easy visualization.
• Allowing step-by-step completion, which can minimize mistakes.
• Providing a reference for comparing changes in value across different parameters.

By completing the table, we solve for \y\ and visualize each computation step. For example, substituting \x = -3\ gives \y = 2\, filling the first row, and so on, consecutively for other \x-values\. Completing this systematically provides clarity and accuracy in the interpretation of equations.

Algebraic substitution is a core technique in solving equations. It involves replacing a variable in an equation with a specified value to simplify and solve the equation. This is crucial in finding particular solutions for equations involving variables.
In our exercise, the equation given is \(y = x + 5\). We substitute specific values for \x\ to find corresponding \y\ values. For example, when \x = -3\, substituting it into the equation results in \(y = -3 + 5 = 2\).
This technique of substitution is beneficial because:

• It allows the exploration of how different values affect the outcome of the equation.
• It is a step-by-step approach, making the process manageable and less prone to error.
• It clarifies the relationship between variables by showing direct computation results.

By using algebraic substitution repeatedly for different x-values in our table, we can systematically solve for \y\ and generate a series of ordered pairs, demonstrating a set of solutions to the equation.