Chapter 3.1, Problem 78HP

To determine

To find: a will sometime, always, or never be consistent and independent with d and explain with reasoning.

Answer to Problem 78HP

$c\text{ and }d$ are consistent and independent, then $c$ is not parallel, then $a$ cannot be parallel to $d$ , so, $a$ must intersect $d$ and must be consistent and independent with $d$ .

Explanation of Solution

Given:

For consistent and dependent grpah there is a same line for both the equations. So, $a\text{ and }b$ are the same line. So, $b$ is parallel to $c$ .

Concept used:

If a system has at least one solution it is said to be consistent.

If a consistent system has exactly one solution, it is independent.

If a consistent system has infinite number of solutions, it is dependent.

When by graphing the equation both equations represent the same line.

Calculation:

For consistent and dependent grpah there is a same line for both the equations. So, $a\text{ and }b$ are the same line. For inconsistent graph there are parallel lines for both the equations. So, $b$ is parallel to $c$ .

For consistent and independent graph there are intersecting lines for both the graph so, $c$ intersect $d$ .

From all the above conclusions $a\text{ and }b$ is the same line. $b$ is parallel to $c$ , so $a$ is also parallel to $c$ , hence, $c\text{ and }d$ are consistent and independent, then $c$ is not parallel, then $a$ cannot be parallel to $d$ , so, $a$ must intersect $d$ and must be consistent and independent with $d$ .