Question

system a
line 1: $y=\frac{3}{4}x + 2$
line 2: $-3x + 4y = 8$
graph of line 1 and line 2
this system of equations is:
consistent dependent
consistent independent
inconsistent
this means the system has:
a unique solution
solution: ( , )
no solution
infinitely many solutions

system b
line 1: $y = 3x + 3$
line 2: $y = 3x - 3$
graph of line 1 and line 2
this system of equations is:
consistent dependent
consistent independent
inconsistent
this means the system has:
a unique solution
solution: ( , )
no solution
infinitely many solutions

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

Explanation:

System A

Step1: Rewrite Line 2

Rewrite \(-3x + 4y = 8\) in slope - intercept form (\(y=mx + b\)).
Add \(3x\) to both sides: \(4y=3x + 8\).
Divide by 4: \(y=\frac{3}{4}x + 2\).

Step2: Compare with Line 1

Line 1 is \(y=\frac{3}{4}x + 2\). Since both lines have the same slope (\(m = \frac{3}{4}\)) and the same y - intercept (\(b = 2\)), they are the same line.

Step3: Determine system type and solution

A system with the same line is consistent dependent. A consistent dependent system has infinitely many solutions.

Step1: Analyze slopes

Line 1: \(y = 3x+3\), slope \(m_1 = 3\).
Line 2: \(y = 3x - 3\), slope \(m_2 = 3\).
The slopes are equal, but the y - intercepts (\(b_1=3\), \(b_2=-3\)) are different. So the lines are parallel.

Step2: Determine system type and solution

A system with parallel lines is inconsistent. An inconsistent system has no solution.

Step1: Analyze slopes

Line 1: \(y=-\frac{1}{2}x - 3\), slope \(m_1=-\frac{1}{2}\).
Line 2: \(y = 2x-3\), slope \(m_2 = 2\).
Since \(m_1
eq m_2\), the lines are not parallel and will intersect at one point.

Step2: Determine system type and solution

A system with intersecting lines is consistent independent. A consistent independent system has a unique solution.

Step3: Find the solution

Set \(-\frac{1}{2}x - 3=2x-3\).
Add \(\frac{1}{2}x\) to both sides: \(-3=\frac{5}{2}x-3\).
Add 3 to both sides: \(0=\frac{5}{2}x\), so \(x = 0\).
Substitute \(x = 0\) into \(y = 2x-3\), we get \(y=-3\).

Answer:

This system of equations is: consistent dependent
This means the system has: infinitely many solutions
Solution: (All points on \(y=\frac{3}{4}x + 2\))

System B