Question

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

Explanation:

System A

Step1: Compare slopes

The equations are \(y = x - 3\) and \(y=x + 1\). The slope \(m\) of both lines (in the form \(y=mx + b\)) is \(m = 1\). Since the slopes are equal but the \(y\) - intercepts (\(-3\) and \(1\)) are different, the lines are parallel.

Step2: Determine system type

Parallel lines never intersect. A system of equations with no intersection point is inconsistent.

System B

Step1: Solve the system

We have the system \(

$$\begin{cases}y=x - 2\\y=-x - 2\end{cases}$$

\). Set \(x-2=-x - 2\). Add \(x\) to both sides: \(x-2+x=-x - 2+x\), \(2x-2=- 2\). Add \(2\) to both sides: \(2x-2 + 2=-2 + 2\), \(2x=0\), so \(x = 0\). Substitute \(x = 0\) into \(y=x - 2\), \(y=-2\).

Step2: Determine system type

Since there is one solution \((0,-2)\), the system is consistent (has at least one solution) and independent (the equations represent different lines).

System C

Step1: Rewrite the second equation

The second equation \(-x + 4y=-4\) can be rewritten in slope - intercept form \(y=\frac{1}{4}x-1\). The first equation is \(y=\frac{1}{4}x-1\).

Step2: Determine system type

Since the two equations represent the same line (same slope \(\frac{1}{4}\) and same \(y\) - intercept \(-1\)), the system is consistent (has solutions) and dependent (the equations are not distinct).

Answer:

• System A: inconsistent, no solution
• System B: consistent independent, a unique solution \((0,-2)\)
• System C: consistent dependent, infinitely many solutions