Question

how many solutions are there to the system of equations ( y = \frac{2}{3}x - 7 ) and ( y = \frac{2}{3}x + 2 )? justify your response.

Explanation:

Step1: Analyze line slopes

Both equations are in slope-intercept form $y=mx+b$, where $m$ is the slope.
For $y=\frac{2}{3}x-7$, slope $m_1=\frac{2}{3}$.
For $y=\frac{2}{3}x+2$, slope $m_2=\frac{2}{3}$.

Step2: Analyze y-intercepts

For $y=\frac{2}{3}x-7$, y-intercept $b_1=-7$.
For $y=\frac{2}{3}x+2$, y-intercept $b_2=2$.

Step3: Determine line relationship

Lines with equal slopes but different y-intercepts are parallel, meaning they never intersect.

Answer:

There are 0 solutions. The two lines have identical slopes ($\frac{2}{3}$) but different y-intercepts, so they are parallel and never cross, resulting in no common solution points.