Question

Classify the system (options: consistent, independent consistent, inconsistent) 4x + 2y = 8y = -2x + 8

Answer 1

Write both equations in standard form:

`ax+by=c`

The first one is already written in standard form:

`4x+2y=8`

To write the second one in standard form, add 2x to both members of the equation:

`\begin{gathered} y=-2x+8 \\ \Rightarrow y+2x=-2x+8+2x \\ \Rightarrow2x+y=8 \end{gathered}`

Notice that if we divide both members of the first equation by 2, we get:

`\begin{gathered} (4x+2y)/(2)=(8)/(2) \\ \Rightarrow2x+y=4 \end{gathered}`

Then, the system of equations is equivalent to:

`\begin{gathered} 2x+y=8 \\ 2x+y=4 \end{gathered}`

For the transitive property of equality, since both left members are the same, we can conclude from that system of equations that 8=4, which is false. Then, the system is inconsistent, since it leads to a contradiction.

Therefore, the answer is:

`\text{Inconsistent}`