Question

what is the correct classification of the system of equations below?
y - 3x = 3
y = 3x - 2
a. parallel
b. coincident
c. intersecting

Explanation:

Step1: Convert to slope - intercept form

For the first equation \(y - 3x=3\), we can rewrite it in slope - intercept form (\(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept) by solving for \(y\).
Adding \(3x\) to both sides of the equation \(y-3x = 3\), we get \(y=3x + 3\).
The second equation is already in slope - intercept form: \(y = 3x-2\).

Step2: Compare slopes and y - intercepts

The slope - intercept form of a line is \(y=mx + b\), where \(m\) is the slope.
For the first line \(y = 3x+3\), the slope \(m_1 = 3\) and the y - intercept \(b_1=3\).
For the second line \(y = 3x - 2\), the slope \(m_2=3\) and the y - intercept \(b_2=- 2\).
Since \(m_1=m_2\) (the slopes are equal) and \(b_1
eq b_2\) (the y - intercepts are different), the two lines are parallel. Parallel lines have the same slope and different y - intercepts, coincident lines have the same slope and same y - intercept, and intersecting lines have different slopes.

Answer:

A. parallel