Question

when solved graphically, which system of equations will have exactly one point of intersection?
\\( y = 0.5x + 30 \\)
\\( y = 0.5x - 30 \\)
\\( y = \frac{3}{5}x + 12 \\)
\\( y = 0.6x - 19 \\)
\\( y = -x - 20 \\)
\\( y = x + 17 \\)
\\( y = -x + 15 \\)
\\( y = -x + 25 \\)
23 multiple choice 2 points
what is the solution set of the following system of equations?
\\( x + y = 7 \\)
\\( x - y = 3 \\)
\\( (10, -3) \\)
\\( (8, -1) \\)
\\( (5, 2) \\)
\\( (3, 4) \\)

Explanation:

First Question (Graphical System Intersection)

Step1: Check slope equality

For a system $y=mx+b$, parallel lines (same $m$) have 0 intersections; different $m$ have 1 intersection.

Step2: Analyze first system

$m_1=0.5, m_2=0.5$ → parallel, 0 intersections.

Step3: Analyze second system

$\frac{3}{5}=0.6$, so $m_1=m_2=0.6$ → parallel, 0 intersections.

Step4: Analyze third system

$m_1=-1, m_2=1$ → different slopes, 1 intersection.

Step5: Analyze fourth system

$m_1=-1, m_2=-1$ → parallel, 0 intersections.

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Second Question (Solution Set of System)

Step1: Add the two equations

$x+y + x-y = 7+3$
$\implies 2x = 10$

Step2: Solve for $x$

$x = \frac{10}{2} = 5$

Step3: Substitute $x=5$ into $x+y=7$

$5 + y = 7$
$\implies y = 7-5=2$

Answer:

$\boldsymbol{y = \frac{3}{5}x + 12 \\ y = 0.6x - 19}$ (second option), $\boldsymbol{y = -x - 20 \\ y = x + 17}$ (third option)