Question

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

system b
line 1: $y = -x - 5$
line 2: $y = -\frac{1}{2}x - 4$
graph of two lines
this system of equations is:
\bigcirc inconsistent
\bigcirc consistent independent
\bigcirc consistent dependent
this means the system has:
\bigcirc a unique solution
solution: (\square, \square)
\bigcirc infinitely many solutions
\bigcirc no solution

Explanation:

System A

Step1: Compare slopes of lines

Line 1: $y=3x+4$, slope $m_1=3$
Line 2: $y=3x-2$, slope $m_2=3$

Step2: Compare y-intercepts

Line 1 intercept: $b_1=4$; Line 2 intercept: $b_2=-2$

Step3: Classify system

Equal slopes, different intercepts = parallel lines, no intersection.

Step1: Compare slopes of lines

Line 1: $y=-x-5$, slope $m_1=-1$
Line 2: $y=-\frac{1}{2}x-4$, slope $m_2=-\frac{1}{2}$

Step2: Solve for intersection

Set equations equal: $-x-5 = -\frac{1}{2}x-4$

Step3: Solve for x

$-x + \frac{1}{2}x = -4 + 5$
$-\frac{1}{2}x = 1$
$x = -2$

Step4: Solve for y

Substitute $x=-2$ into $y=-x-5$:
$y = -(-2)-5 = 2-5 = -3$

Answer:

This system of equations is: $\boldsymbol{\text{inconsistent}}$
This means the system has: $\boldsymbol{\text{no solution}}$

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System B