Answer:
Part 1) option D) No solutions
Part 2) option C \(y=9\)
Part 3) option E \(E(0,-3)\)
Step-by-step explanation:
Part 1) we have
\(3x-2y=6\) -------> equation A
\(6x-4y=14\) -------> equation B
Divide the equation B by \(2\) both sides
\(6x-4y=14\) --------> \(3x-2y=7\)
Equation A and equation B represent a parallel lines, because their have the same slope \(m=\frac{3}{2}\)
therefore
The system of equation does not have solution
Part 2) we have
\(5x+4y=1\) -------> equation A
\(4x+3y=-1\) -------> equation B
Multiply the equation A by \(4\) both sides
\(5x+4y=1\) -------> \(20x+16y=4\) -------> equation C
Multiply the equation B by \(-5\) both sides
\(4x+3y=-1\) ------> \(-20x-15y=5\) ------> equation D
Adds equation C and equation D
\(20x+16y=4\\-20x-15y=5\\---------\\16y-15y=4+5\\ y=9\)
Part 3) we have
\(y<2x+4\) -------> inequality A
\(y<-2x+2\) -------> inequality B
we know that
If a point lie in the solution set of the system of inequalities, then the point must be satisfy the inequalities of the system
a) point \(C(1,0)\)
Verify inequality A
\(0<2(1)+4\)
\(0<6\) -------> is true
Verify inequality B
\(0<-2(1)+2\)
\(0<0\) -----> is not true
therefore
the point \(C(1,0)\) does not lie in the solution set of the system of inequalities
b) point \(D(-5,-2)\)
Verify inequality A
\(-2<2(-5)+4\)
\(-2<-6\) -------> is not true
therefore
the point \(D(-5,-2)\) does not lie in the solution set of the system of inequalities
c) point \(E(0,-3)\)
Verify inequality A
\(-3<2(0)+4\)
\(-3<4\) -------> is true
Verify inequality B
\(-3<-2(0)+2\)
\(-3<2\) -----> is true
therefore
the point \(E(0,-3)\) lies in the solution set of the system of inequalities
d) point \(F(-1,5)\)
Verify inequality A
\(5<2(-1)+4\)
\(5<2\) -------> is not true
therefore
the point \(F(-1,5)\) does not lie in the solution set of the system of inequalities
