Question

two systems of equations are given below. for each system, choose the best description of its solution. if applicable, give the solution.
system a
4x+5y=9
4x+5y=4
\t\t\t\t\t\t\t\t○ the system has no solution.
\t\t\t\t\t\t\t\t○ the system has a unique solution:
\t\t\t\t\t\t\t\t\t\t(x,y)= (□,□)
\t\t\t\t\t\t\t\t○ the system has infinitely many solutions.
system b
\ty=7x
\ty=3x
\t\t\t\t\t\t\t\t○ the system has no solution.
\t\t\t\t\t\t\t\t○ the system has a unique solution:
\t\t\t\t\t\t\t\t\t\t(x,y)= (□,□)
\t\t\t\t\t\t\t\t○ the system has infinitely many solutions.

Explanation:

Step1: Analyze System A

We have the equations \(4x + 5y=9\) and \(4x + 5y = 4\). Let's subtract the second equation from the first: \((4x + 5y)-(4x + 5y)=9 - 4\), which simplifies to \(0=5\), a contradiction. So System A has no solution.

Step2: Analyze System B

We have \(y = 7x\) and \(y=3x\). Set them equal: \(7x=3x\), subtract \(3x\) from both sides: \(4x = 0\), so \(x = 0\). Then \(y=7(0)=0\) (or \(y = 3(0)=0\)). So the solution is \((0,0)\), a unique solution.

Answer:

System A: The system has no solution.
System B: The system has a unique solution: \((x,y)=(0,0)\)