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In inverse proportionality, the relationship between two variables is one where one quantity increases and the other decreases at a rate that keeps their product constant. This constant is known as the Proportionality Constant. For two variables, \(x\) and \(y\), to be inversely proportional, their product must equal a constant value \(k\). So, the relationship is expressed as:

• \(xy = k\)

To find the proportionality constant \(k\), you need a pair of values for \(x\) and \(y\). In the exercise you provided, when \(x = 3\) and \(y = 4\), \(k\) is calculated as \(3 \times 4 = 12\). This means that no matter the values of \(x\) and \(y\), as long as they are inversely proportional, their product will always be 12.

Coordinates are used to describe the position of a point on a plane. Each point is represented by an ordered pair \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position. When solving problems involving inverse proportionality, you often need to find missing coordinates based on the relationship between \(x\) and \(y\) given the proportionality constant \(k\).In this exercise, several coordinate pairs are partially given, and our task is to find the missing values using the inverse proportionality equation \(xy = 12\). With each scenario, either \(x\) or \(y\) is given, and the other can be found by substituting into the equation and solving for the unknown.

• For example, in the coordinate \((4, y)\), plug \(x = 4\) into \(xy = 12\) to solve for \(y\).
• If the coordinate \((x, 2)\) is given, use \(y = 2\) to determine \(x\).

Algebraic equations involve mathematical expressions with variables that can be solved to find unknown values. In the case of inverse proportionality, the core algebraic equation is \(xy = k\). This equation is manipulated to find missing values of \(x\) or \(y\) when one of them is known.To solve these equations:
1. Identify which variable is missing.2. Substitute the known values into the equation.3. Rearrange the equation to solve for the unknown variable.
For example:

• If given \((4, y)\), substitute \(x = 4\) into the equation \(4y = 12\) and solve for \(y\) by dividing both sides by 4.
• For \((x, 2)\), replace \(y\) with 2 in \(xy = 12\) to get \(x \times 2 = 12\). Solve for \(x\) by dividing both sides by 2.

Each step involves basic algebraic manipulation to isolate the unknown variable, which helps in finding its value accurately. This approach is fundamental in solving inverse proportionality problems.