How many solutions does this system have

How many solutions does this system have? Find all solutions to the system of equations.ÂÂ

3x+2y=-1

5x+4y=0

The correct answer and explanation is :

We are given the following system of linear equations:

1. ( 3x + 2y = -1 )
2. ( 5x + 4y = 0 )

Step 1: Solve the system using substitution or elimination

We will use the elimination method here, where we aim to eliminate one of the variables by manipulating the equations.

First, let’s eliminate ( y ) by multiplying the first equation by 2:

[
2(3x + 2y) = 2(-1)
]
[
6x + 4y = -2
]

Now, we have the modified system of equations:

1. ( 6x + 4y = -2 )
2. ( 5x + 4y = 0 )

Next, subtract the second equation from the first equation:

[
(6x + 4y) – (5x + 4y) = -2 – 0
]
[
6x – 5x = -2
]
[
x = -2
]

Step 2: Substitute ( x = -2 ) into one of the original equations

Now that we know ( x = -2 ), we can substitute this value back into one of the original equations to solve for ( y ). Let’s substitute into the first equation:

[
3x + 2y = -1
]
Substitute ( x = -2 ):
[
3(-2) + 2y = -1
]
[
-6 + 2y = -1
]
[
2y = 5
]
[
y = \frac{5}{2}
]

Step 3: Conclusion

Thus, the solution to the system of equations is:
[
x = -2, \quad y = \frac{5}{2}
]

The system has exactly one solution. The solution is the ordered pair ( \left( -2, \frac{5}{2} \right) ).

Explanation of the Process

The system of equations is a pair of linear equations in two variables. To solve such systems, we can use several methods: substitution, elimination, or graphing. In this case, we opted for the elimination method. We manipulated the equations to eliminate one variable and solve for the other. After solving for ( x ), we substituted that value back into one of the original equations to find ( y ). The final solution shows that the system has one unique solution, which means the lines represented by the two equations intersect at a single point on the coordinate plane.