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 $2x + 3y = 9$ $2x + 3y = 4$ $\bigcirc$ the system has no solution. $\bigcirc$ the system has a unique solution: $(x, y) = (\square, \square)$ $\bigcirc$ the system has infinitely many solutions. system b $y = 7x$ $y = 3x$ $\bigcirc$ the system has no solution. $\bigcirc$ the system has a unique solution: $(x, y) = (\square, \square)$ $\bigcirc$ the system has infinitely many solutions.

Explanation:

System A

Step1: Analyze the two equations

We have the system:
$$2x + 3y = 9$$
$$2x + 3y = 4$$
If we subtract the second equation from the first equation, we get:
$$(2x + 3y)-(2x + 3y)=9 - 4$$
$$0=5$$
This is a contradiction, which means there is no solution that satisfies both equations simultaneously.

Step1: Set the two equations equal

We have \(y = 7x\) and \(y=3x\). Since both equal \(y\), we can set them equal to each other:
$$7x=3x$$

Step2: Solve for \(x\)

Subtract \(3x\) from both sides:
$$7x - 3x=3x - 3x$$
$$4x = 0$$
Divide both sides by 4:
$$x = 0$$

Step3: Find \(y\)

Substitute \(x = 0\) into \(y = 7x\) (we could also use \(y = 3x\)):
$$y=7\times0 = 0$$

Answer:

The system has no solution.

System B