91Ó°ÊÓ

In mathematics, an ordered pair is a collection of two elements usually written in the form \((x, y)\). These pairs are specifically designated to represent a specific point on a coordinate plane. It's "ordered" in the sense that \((x, y)\) is not the same as \((y, x)\). The first element typically represents the horizontal component (x-coordinate), and the second represents the vertical component (y-coordinate). This ordering is crucial, especially if we are plotting the point on a graph.
For instance, in this exercise, the ordered pair \((12, ?)\) implies a point on the coordinate plane where the x-value, or horizontal position, is known to be 12. The question is about determining the y-value to complete the pair, which, when combined, pinpoints a precise spot on a graph.

Linear equations are a fundamental part of algebra and are expressed in the form \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, or where the line crosses the y-axis. Linear equations graph straight lines and have consistent slopes, meaning they change at a constant rate.
In our example, the linear equation is given as \(y = \frac{2}{3}x + 5\).

• The slope \(\frac{2}{3}\) indicates that for every increase of 3 units in x, y will increase by 2 units.
• The y-intercept 5 tells us that if x were to be 0, y would be 5.

Understanding how to manipulate these equations is crucial for predicting the behavior of real-world phenomena described by such relationships.

The substitution method is a technique used in solving linear equations, particularly when working with ordered pairs. The basic principle involves substituting one known value into an equation to solve for another unknown variable.
For this particular task, we take the given x-value from the ordered pair \((12, ?)\) and substitute it into the equation \(y = \frac{2}{3}x + 5\) to find y. Here's a brief breakdown of the steps:

• Step 1: The equation provided is \(y = \frac{2}{3}x + 5\).
• Step 2: Substitute the given x-value (12) into the equation to solve for y.
• Step 3: Simplify the equation. Start by computing \(\frac{2}{3} \times 12 = 8\).
• Step 4: Then add 5 to the result, leading us to \( y = 13 \).

By following these steps, the goal is to find the unknown y-value making our ordered pair complete as \((12, 13)\), allowing us to accurately plot or use this point in further calculations.