Determine whether the following system is consistent or inconsjstent. If the system is consistent, determine whether the equations are dependent or independent. Do not solve the system.
\[
\left\{\begin{array}{l}
y=-4 x+5 \\
y=5 x+7
\end{array}\right.
\]
Choose the correct answer below.
A. The given system is consistent and independent.
B. The given system is inconsistent.
c. The given system is consistent and dependent.
\(\boxed{\text{The given system is consistent and independent.}}\)
Step 1 :The given system of equations is: \(y = -4x + 5\) and \(y = 5x + 7\)
Step 2 :To determine whether the system is consistent or inconsistent, we need to check if there is at least one solution that satisfies both equations.
Step 3 :Setting the two equations equal to each other, we get: \(-4x + 5 = 5x + 7\)
Step 4 :Solving for x, we get: \(9x = -2\) which simplifies to \(x = -\frac{2}{9}\)
Step 5 :Substituting \(x = -\frac{2}{9}\) into the first equation, we get: \(y = -4(-\frac{2}{9}) + 5 = \frac{8}{9} + 5 = \frac{53}{9}\)
Step 6 :Substituting \(x = -\frac{2}{9}\) into the second equation, we get: \(y = 5(-\frac{2}{9}) + 7 = -\frac{10}{9} + 7 = \frac{53}{9}\)
Step 7 :Since the values of y are the same for both equations, the system is consistent.
Step 8 :To determine whether the equations are dependent or independent, we need to check if one equation can be obtained from the other by multiplication or addition.
Step 9 :In this case, the two equations cannot be obtained from each other by multiplication or addition, so the equations are independent.
Step 10 :\(\boxed{\text{The given system is consistent and independent.}}\)