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The constant of proportionality (often denoted as \(k\)) is a key concept when dealing with direct proportionality. It is the factor that remains consistent in a relationship where two variables change proportionally. For instance, if \(y\) changes directly in relation to \(x\), then \(y = kx\) holds true. The constant \(k\) quantifies how much \(y\) changes for a given change in \(x\).

In the exercise provided, we see that when \(y = 13\) and \(x = 10\), the constant of proportionality \(k\) is determined by the equation \(13 = k \times 10\). Solving for \(k\), we find \(k = 1.3\). This tells us that for every unit change in \(x\), \(y\) changes by 1.3 units.

A linear relationship in mathematics is one where there is a direct proportional connection between two variables.
This means if you plot these variables on a graph, the result will be a straight line. This line's steepness, or slope, represents the constant of proportionality.

In the equation \(y = kx\), both \(y\) and \(x\) change at a constant rate determined by \(k\). So, if you have a change in \(x\) of, say, 1 unit, the change in \(y\) will be k units. This consistent relationship between \(x\) and \(y\) is what characterizes a linear relationship.

• Remember: In a linear graph, the line does not curve.
• The slope \(m\) of the line in the equation \(y = mx + b\) (where \(b = 0\) in direct proportionality) represents the constant of proportionality.

For our specific example, \(y = 1.3x\) forms a linear relationship because the growth of \(y\) and \(x\) is steady and predictable, described accurately by the slope \(k = 1.3\).

Proportional equations are expressions that maintain a definitive ratio between their variables. In a direct proportional equation, this ratio is defined by the constant of proportionality \(k\).

The standard form of a proportional equation is \(y = kx\). This equation allows you to predict the value of \(y\) for any value of \(x\), as long as the condition of proportionality holds.

In the exercise, we first identified the constant \(k\) as 1.3. With this established, we used the equation \(y = 1.3x\) to find \(y\) when \(x = 2.5\). By substituting 2.5 into the equation, we calculated \(y\), which came out to be 3.25.

• Proportional equations help streamline predictions in directly proportional scenarios.
• They solidify the understanding that for a set ratio, both variables work in tandem, reflecting the essence of direct proportionality.

This powerful relationship and its resulting equations thus enable easy calculations and model numerous real-world phenomena effectively.