91Ó°ÊÓ

Linear functions are a fundamental concept in algebra and calculus. They are expressions that create a straight line when plotted on a graph. This makes them one of the simplest forms we can work with in mathematics.
The general formula for a linear function in two dimensions is given by: - \( f(x, y) = ax + by + c \),where:

• \( a \) and \( b \) are coefficients that determine the direction of the line.
• \( c \) is the constant term, acting as the y-intercept when the function is rearranged in slope-intercept form \( y = mx + b \).

In this context, we are focusing on linear functions in the form \( ax + by = c \) where the function is usually rearranged to find values of \( y \) based on \( x \), which helps in plotting them on a graph. Understanding linear functions is crucial as they serve as a foundation for more complex mathematical concepts.

Parallel lines are lines on a graph that never meet. They have the same slope but different y-intercepts. When we talk about parallel lines in the context of the exercise, each level curve created by a linear function with a different \( c \) value forms a parallel line.
For example, a level curve represented by \( 3x - 2y = c \) for various \( c \) values (such as -2, 0, and 2) will yield parallel lines because after rearranging to \( y = mx + b \), we always have the same slope (\( m = \frac{3}{2} \)).

• The slope, \( \frac{3}{2} \) in this case, indicates how steep the lines are when graphed.
• The y-intercept (the constant \( k \) in \( y = \frac{3}{2}x + k \)) determines where the line crosses the y-axis and depends on the value of \( c \).

In conclusion, knowing how to identify and work with parallel lines is essential, especially in understanding the graphical representation of equations such as level curves.

Visualizing mathematical equations on a graph helps in understanding their behavior and characteristics. For the given function \( f(x, y) = 3x - 2y \), graphing its level curves means plotting the lines resulting from different \( c \) values, namely -2, 0, and 2.
To graph these equations, follow these simple steps:

• Start by rearranging each equation into slope-intercept form (\( y = mx + b \)).
• Create a table of values for chosen \( x \) inputs to calculate corresponding \( y \) values.
• Plot these points on the Cartesian plane and connect them to form the line.
• Repeat this process to include all level curves (lines) on the same graph.

For example:- For \( c = -2 \), the line is \( y = \frac{3}{2}x + 1 \).- For \( c = 0 \), the line is \( y = \frac{3}{2}x \).- For \( c = 2 \), the line is \( y = \frac{3}{2}x - 1 \).
When graphing, it's important to label each line according to its respective \( c \) value to avoid confusion. Graphing such equations not only helps in visualizing the solution but also provides insight into their parallel nature.