Question

system c
line 1: $y=\frac{2}{3}x + 1$
line 2: $-2x + 3y = 3$
graph of two lines l1 and l2 on a coordinate plane
this system of equations is:
∘ inconsistent
∘ consistent dependent
∘ consistent independent
this means the system has:
∘ a unique solution
solution: ( , )
∘ no solution
∘ infinitely many solutions

Explanation:

Step1: Rewrite Line 2 in slope - intercept form

We start with the equation of Line 2: \(-2x + 3y=3\). We want to solve for \(y\) to get it in the form \(y = mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).
First, add \(2x\) to both sides of the equation: \(3y=2x + 3\).
Then, divide each term by 3: \(y=\frac{2}{3}x + 1\).

Step2: Compare the two lines

The equation of Line 1 is \(y=\frac{2}{3}x + 1\) and the equation of Line 2 (after rewriting) is also \(y=\frac{2}{3}x + 1\). This means that the two lines are identical (they have the same slope and the same y - intercept).

Step3: Determine the type of system

• An inconsistent system has no solutions (the lines are parallel, same slope different y - intercepts).
• A consistent dependent system has infinitely many solutions (the lines are the same, coincident).
• A consistent independent system has a unique solution (the lines intersect at one point, different slopes).

Since the two lines are the same, the system is consistent dependent. A consistent dependent system has infinitely many solutions.

Answer:

This system of equations is: consistent dependent
This means the system has: infinitely many solutions