graph the following system of equations and choose the correct statement. \\(\\begin{cases} y =…

Explanation:

Step1: Analyze the first equation ( y = -\frac{2}{3}x - 6 )

This is a linear equation in slope - intercept form ( y=mx + b ), where the slope ( m=-\frac{2}{3} ) and the y - intercept ( b=-6 ). To find two points on this line, we can use the y - intercept and another point. When ( x = 0 ), ( y=-6 ). When ( y = 0 ), we solve ( 0=-\frac{2}{3}x-6 ), adding 6 to both sides gives ( 6=-\frac{2}{3}x ), and then multiplying both sides by ( -\frac{3}{2} ) gives ( x=-9 ). So two points on this line are ( (0, - 6) ) and ( (-9,0) ).

Step2: Analyze the second equation ( y=\frac{2}{3}x - 1 )

This is also a linear equation in slope - intercept form with slope ( m = \frac{2}{3} ) and y - intercept ( b=-1 ). When ( x = 0 ), ( y=-1 ). When ( y = 0 ), we solve ( 0=\frac{2}{3}x-1 ), adding 1 to both sides gives ( 1=\frac{2}{3}x ), and then multiplying both sides by ( \frac{3}{2} ) gives ( x=\frac{3}{2}=1.5 ). So two points on this line are ( (0,-1) ) and ( (1.5,0) ).

Step3: Find the intersection point (solution of the system)

To find the intersection of the two lines, we set the two equations equal to each other: ( -\frac{2}{3}x - 6=\frac{2}{3}x - 1 ) Add ( \frac{2}{3}x ) to both sides: ( - 6=\frac{4}{3}x-1 ) Add 1 to both sides: ( -5=\frac{4}{3}x ) Multiply both sides by ( \frac{3}{4} ): ( x=-\frac{15}{4}=-3.75 ) Substitute ( x = -\frac{15}{4} ) into the second equation ( y=\frac{2}{3}x - 1 ): ( y=\frac{2}{3}\times(-\frac{15}{4})-1=\frac{-30}{12}-1=-\frac{5}{2}-1=-\frac{7}{2}=-3.5 ) So the intersection point is ( (-\frac{15}{4},-\frac{7}{2}) ) or ( (-3.75,-3.5) )

Step4: Analyze the slopes

The slope of the first line ( m_1 = -\frac{2}{3} ) and the slope of the second line ( m_2=\frac{2}{3} ). Since ( m_1\neq m_2 ) and ( m_1\times m_2=-\frac{2}{3}\times\frac{2}{3}=-\frac{4}{9}\neq - 1 ), the two lines are not parallel and not perpendicular. They intersect at exactly one point, which means the system has a unique solution.

Answer:

The system of equations ( \left{\begin{array}{l}y = -\frac{2}{3}x - 6\y=\frac{2}{3}x - 1\end{array}\right. ) has a unique solution (the two lines intersect at one point). The intersection point is ( \left(-\frac{15}{4},-\frac{7}{2}\right) ) (or ( (-3.75,-3.5) )).