Question

Describe the family of } & \text { planes }\end{array}\( represented by the equation, where \)c$ is any real number. $$ x+y=c $$

Short answer

The family of planes represented by the equation \(x + y = c\) is a set of lines in a 2-dimensional plane. Each line in this family is parallel to each other and has a slope of -1. The y-intercept of each line changes with different values of \(c\), adapting the line's position in an upward or downward direction in the y-axis.

Step-by-step solution

Understanding the equation

The equation given is \(x + y = c\). This is a linear equation in two variables (x and y), defining a line in a two-dimensional plane. Since \(c\) can be any real number, the equation represents not just one line, but a family of lines.

Describing the family of planes

Each specific value of \(c\) yields a different line in the 2D plane. When \(c\) increases, the line moves upwards in the y-axis direction. When \(c\) decreases, the line moves downwards. So, effectively, changing \(c\) changes the y-intercept of the line, while the slope remains the same. All lines have a slope of -1.

Visual Representation

For a better understanding, it's beneficial to draw the lines on a graph using various values for \(c\). This gives a visual representation of how the family of lines moves in the plane.

Key concepts

Linear Equations

When dealing with algebra, one of the fundamental units you'll encounter is the linear equation. This is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The most basic form of a linear equation with two variables, x and y, is given by the formula \(ax + by = c\).

The equation \(x + y = c\) from our exercise exemplifies a linear equation with two variables. Here, 'a' and 'b' are both 1, and 'c' can be any real number. This simplicity allows us to understand that for different values of 'c', we get distinct lines on the graph, but all will follow the straight line pattern. Every line in this family will have the same general shape and direction, but they will be positioned differently depending on the value of 'c'.

Understanding linear equations like this forms the backbone for more complex mathematical problems, such as systems of equations or linear programming. By mastering their behavior, by recognizing that changing 'c' alters the y-intercept, we deepen our understanding of how equations reflect graphical representations.

Two-dimensional Plane

The two-dimensional plane, often referred to as 2D, is the flat surface that extends infinitely in two directions. It's where the action happens for linear equations and many other geometrical concepts. When you plot the linear equation \(x + y = c\), each value of 'c' gives us a different line on this plane, representing a distinct relationship between x and y.

In our textbook exercise, the 2D plane serves as a stage to display the family of lines created by varying 'c'. If you imagine each line as a tightrope strung across this plane, by changing 'c' you're essentially moving each rope up or down, creating a series of parallel lines, since all lines share the same slope. This visual can help to concretize the abstract concept of a 'family of lines' and how linear equations can map out not just single lines, but an entire family within the confines of a 2D plane.

Slope of a Line

The slope of a line is a measure of its steepness or incline, defined as the ratio of the rise (the change in y) to the run (the change in x). For the equation of a line in the form \(y = mx + b\), 'm' represents the slope. It tells us how much y changes for each increase in x.

In our exercise, the slope of the family of lines represented by \(x + y = c\) is -1. No matter what value 'c' takes, the amount that y decreases for every one-unit increase in x is consistent across all lines. It's like having a set of stairs where each step down across the plane is equal; every step down in y is a step to the right in x. This constant slope results in parallel lines since each line tilts at the same angle. The concept of slope is paramount when learning about lines, as it determines the direction and angle at which they traverse the plane. Understanding that even though we have different lines for each 'c', the slope remaining constant implies that these lines never intersect and maintain parallelism - an intriguing characteristic of the family of lines we're studying.