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The point-slope form of a linear equation is a useful way to express the equation of a line. It's represented by the formula \( y - y_1 = m(x - x_1) \), where:

• \( m \) is the slope of the line.
• \((x_1, y_1)\) is a point through which the line passes.

When dealing with real-world problems, the point-slope form is handy for quickly writing the equation of a line when you know one point on the line and the slope. For example, in the context of the Coffee Stop exercise, if we have a point \((33, 49.75)\), and a slope \( m = 0.75 \), we substitute these values into the formula to get the equation: \( y - 49.75 = 0.75(x - 33) \).
This allows us to model the relationship between the number of drinks purchased and the total cost easily.

The \( y \)-intercept is a key element of linear equations, particularly in slope-intercept form, which is \( y = mx + b \). Here, the \( y \)-intercept \( b \) is the value of \( y \) when \( x = 0 \). In other words, it is the point where the line crosses the \( y \)-axis.
In practical terms, the \( y \)-intercept often represents a base or starting amount in real-world scenarios. In the Coffee Stop example, the \( y \)-intercept is \( 25 \), which corresponds to the initial cost of purchasing the mug. This is the total cost when no additional drinks have been bought. Thus, understanding the \( y \)-intercept helps in interpreting the data and in making sense of the relationship expressed by the linear equation.

Linear equations are mathematical expressions used to describe a straight line on a graph. These equations take the form \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) is the slope of the line.
• \( b \) is the \( y \)-intercept.

Linear equations are crucial for modeling relationships between variables in various contexts, such as physics, economics, and everyday situations. In the example provided, the linear equation \( y = 0.75x + 25 \) models the total cost \( y \) as a function of the number of hot drinks \( x \). This equation effectively communicates how the cost increases with the number of drinks purchased, starting from the initial fixed cost of the mug.

A cost model is a mathematical representation of how costs change with varying levels of activity or other factors. In the case of the Coffee Stop, the cost model is represented by the linear equation derived from understanding the problem. This equation, \( y = 0.75x + 25 \), reflects how the total cost \( y \) depends on the number of hot drinks \( x \) purchased.
Key features of this cost model include:

• The fixed cost component, \( 25 \), representing the mug's purchase price.
• The variable cost component, \( 0.75 \), indicating how much each additional hot drink adds to the total cost.

This model is highly effective for predicting total costs in similar situations, providing insights into both the initial expense and variable costs associated with purchasing additional items.