Question

Match each system on the left with all words that describe the system on the right.Choices on the right can be used more than once.y = 2x + 3x + y = - 33y = 9x – 62y - 6x = 4y = - 2x + 2x + 2y = 4Are the systems inconsistent, consistent, independent or dependent?

Answer 1

We have to find if the system of equations are inconsistent, consistent, independent or dependent.

If a system of linear equations has at least one solution, it is consistent.

A consistent system can be independent if it has only one solution or dependent if it has infinite many solutions.

If it does not have any solutions, the system is inconsistent.

We can find which type of system we have by looking at this (applied to 2 unknowns systems, but it can be extended to higher dimensions):

1) If two of the equations are linear combinations of each other, they are the same line, so the system has infinite many solutions. The system is consistent and dependent.

2) If two of the equations have the same slope but are not the same line, they don't intersect, so the system does not have a solution. The system is then inconsistent.

3) If none of the above situations is the case for the system, the system is consistent and independent.

We start with the first system:

`\begin{gathered} y=2x+3 \\ x+y=-3 \end{gathered}`

We can rewrite this as:

`\begin{gathered} y=2x+3 \\ y=-x-3 \end{gathered}`

As the lines are not parallel, we have a system that is consistent and independent.

The second system is:

`\begin{gathered} 3y=9x-6 \\ 2y-6x=4 \end{gathered}`

We can rearrange the second equation by multiplying both sides by 3/2 so we can compare the two equations with the same coefficient for y:

`\begin{gathered} 2y-6x=4 \\ (3)/(2)(2y-6x)=(3)/(2)(4) \\ (3\cdot2)/(2)y-(6\cdot3)/(2)x=(3\cdot4)/(2) \\ 3y-9x=6 \\ 3y=9x+6 \end{gathered}`

The lines have the same slope, as the coefficients of x and y are the same in both equations, but they are not the same line (the independent term differs), so they are parallel lines.

The system is then inconsistent.

The third system is:

`\begin{gathered} y=-2x+2 \\ x+2y=4\longrightarrow y=-(1)/(2)x+2 \end{gathered}`

The lines are not parallel, so the system is consistent and independent.

Answer:

a) Consistent and independent

b) Inconsistent

c) Consistent and independent