Question

line 1: $y = \frac{1}{3}x$
line 2: $y = \frac{1}{3}x + 1$
this system of equations is:
consistent independent
consistent dependent
inconsistent
this means the system has:
a unique solution
solution:
infinitely many solutions
no solution

Explanation:

Step 1: Analyze the slopes of the two lines

The general form of a linear equation is \(y = mx + b\), where \(m\) is the slope.
For Line 1: \(y=\frac{1}{3}x\), the slope \(m_1=\frac{1}{3}\)
For Line 2: \(y=\frac{1}{3}x + 1\), the slope \(m_2=\frac{1}{3}\)
Since \(m_1=m_2=\frac{1}{3}\), the lines are parallel.

Step 2: Analyze the y - intercepts

For Line 1: \(y=\frac{1}{3}x\), the y - intercept \(b_1 = 0\)
For Line 2: \(y=\frac{1}{3}x+1\), the y - intercept \(b_2=1\)
Since \(b_1
eq b_2\), the lines are not coincident.

Answer:

The system of equations is: inconsistent.
This means the system has: no solution.