Question

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how many solutions does the system of equations below have?
2y = 2x - 6
x + 3y + 15 = 0
no solution
one solution
infinitely many solutions

Explanation:

Step1: Rewrite first equation

Rewrite $2y = 2x - 6$ in slope - intercept form $y=mx + b$. Divide by 2: $y=x - 3$, so slope $m_1 = 1$ and y - intercept $b_1=-3$.

Step2: Rewrite second equation

Rewrite $x + 3y+15 = 0$ in slope - intercept form. First, isolate $y$: $3y=-x - 15$, then $y=-\frac{1}{3}x-5$. So slope $m_2 =-\frac{1}{3}$ and y - intercept $b_2 = - 5$.

Step3: Analyze slopes

Since $m_1
eq m_2$ (where $m_1 = 1$ and $m_2=-\frac{1}{3}$), the two lines intersect at a single point.

Answer:

one solution