Given the system of equations:
[tex]\begin{gathered} y=\frac{1}{3}x+4 \\ \\ y=-3x+2 \end{gathered}[/tex]
Ley's determine if the system is inconsistent or consistent and dependent or independent.
Let's first solve the system of equations.
Eliminate the equivalent sides and combine the equations.
We have:
[tex]\begin{gathered} \frac{1}{3}x+4=-3x+2 \\ \\ \frac{x}{3}+3x=2-4 \end{gathered}[/tex]
Solving further:
[tex]\begin{gathered} \frac{x+9x}{3}=-2 \\ \\ \frac{10x}{3}=-2 \\ \\ Multiply\text{ both sides by 3:} \\ \frac{10x}{3}*3=-2*3 \\ \\ 10x=-6 \\ \\ Divide\text{ both sides by 10:} \\ \frac{10x}{10}=-\frac{6}{10} \\ \\ x=-\frac{3}{5} \end{gathered}[/tex]
Now, plug in -3/5 for x in any of the equations:
[tex]\begin{gathered} y=\frac{1}{3}x+4 \\ \\ y=\frac{1}{3}*(-\frac{3}{5})+4 \\ \\ y=-\frac{1}{5}+4 \\ \\ y=\frac{-1+20}{5} \\ \\ y=\frac{19}{5} \end{gathered}[/tex]
Therefore, we have the solutions:
[tex](x,y)==>(-\frac{3}{5},\frac{19}{5})[/tex]
The system is consistent and independent since it has a definite solution.
The system has just one solution, so we can say it is consistent and independent.
ANSWER:
Consistent and independent.
