Question

The equation \(2 x-y=C\) defines a family of lines, one line for each value of \(C\). On one set of coordinate axes, graph the members of the family when \(C=-4, C=0,\) and \(C=2\) Can you draw a conclusion from the graph about each member of the family?

Short answer

All lines are parallel with slope 2 but shifted vertically based on \(C\).

Step-by-step solution

- Understanding the Equation

The given equation is in the form of a linear equation: \(2x - y = C\). You will graph this equation for different values of \(C\).

- Substitute and Simplify for \(C=-4\)

Substitute \(C = -4\) into the equation: \(2x - y = -4\). Rearrange it to the slope-intercept form \(y = mx + b\): \(y = 2x + 4\). This is the first equation to graph.

- Substitute and Simplify for \(C=0\)

Substitute \(C = 0\) into the equation: \(2x - y = 0\). Rearrange it: \(y = 2x\). This is the second equation to graph.

- Substitute and Simplify for \(C=2\)

Substitute \(C = 2\) into the equation: \(2x - y = 2\). Rearrange it: \(y = 2x - 2\). This is the third equation to graph.

- Plot the Graphs

Draw the coordinate axes. For each line, plot points by choosing values for \(x\) and solving for \(y\). For \(C = -4\) (\(y = 2x + 4\)), example points could be \((0, 4)\) and \((1, 6)\). For \(C = 0\) (\(y = 2x\)), example points could be \((0,0)\) and \((1, 2)\). For \(C = 2\) (\(y = 2x - 2\)), example points could be \((0, -2)\) and \((1, 0)\).

- Analyze the Graphs

Observe that all lines are parallel since the slope (\(m = 2\)) is the same for all equations. The lines are shifted vertically depending on the value of \(C\).

Key concepts

Family of Lines

A family of lines is a set of linear equations that differ only by their constant term. For example, consider the equation given:

\(2x - y = C\).

Each unique value of \(C\) gives us a different line, but these lines have something in common: their slope.

This means if you graph these lines, they'll all have the same angle of incline but will be spaced differently on the graph. These lines are parallel to each other.

The concept of a family of lines is important for visualizing how changes in specific parameters (like \(C\)) affect the entire set of solutions.

Slope-Intercept Form

The slope-intercept form of a linear equation is represented as:

\(y = mx + b\).

In this form, \(m\) represents the slope, and \(b\) represents the y-intercept (the point where the line crosses the y-axis).

For the given equation: \(2x - y = C\)

we rearrange it to the slope-intercept form. For instance, when \(C = -4\), the equation becomes:

\(y = 2x + 4\).

Here, the slope \(m\) is 2, and the y-intercept \(b\) is 4. All lines in the given family have the same slope of 2, indicating they are parallel.

Parallel Lines

Lines are said to be parallel if they have the same slope. This means they will never intersect no matter how far they are extended.

In our equation: \(2x - y = C\)

for different values of \(C\) (like \(-4, 0, 2\)), the slope remains constant at 2. As a result, all these lines are parallel to each other.

This is a clear property of a family of lines as the main difference between the lines is the y-intercept—how high or low the line is on the graph.

Coordinate System

The coordinate system is essential for graphing lines. It consists of an x-axis (horizontal) and a y-axis (vertical).

In this system, each point is represented by a pair of numbers (x, y). To graph a line, you need to plot points that satisfy the equation and connect them with a straight line.

For instance, to graph the line \(y = 2x + 4\) when \(C = -4\), you would plot points like (0, 4) and (1, 6) and then draw a line through them.

Understanding how to navigate this system helps you visualize different mathematical concepts effectively.