How many solutions does the following system have?
2x+3y=1
-3x-2y=-1
%0D%0A%0D%0A %0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0Ainfinitely many solutions%0D%0Ainfinitely many solutions%0D%0A%0D%0Atwo solutions%0D%0Atwo solutions%0D%0A%0D%0Aone solution%0D%0Aone solution%0D%0A%0D%0Ano solutions
The given system of equations is:
2x + 3y = 1
-3x - 2y = -1
To find the number of solutions, we can use the concept of determinants. If the determinant of the coefficient matrix is non-zero, the system has one unique solution. If the determinant is zero, the system can either have infinitely many solutions or no solutions.
The determinant of the coefficient matrix is:
| 2 3 |
| -3 -2 |
= (2 * -2) - (3 * -3)
= 4 + 9
= 13
Since the determinant is non-zero (i.e., 13 is not zero), the system has one unique solution.
Therefore, the correct answer is: one solution.
To find the number of solutions in the given system of equations, we can use the concept of determinants.
Step 1: Write down the given equations:
2x + 3y = 1 ...(Equation 1)
-3x - 2y = -1 ...(Equation 2)
Step 2: Set up the coefficient matrix:
The coefficient matrix is formed by taking the coefficients of the variables in the equations. In this case, the coefficient matrix is:
| 2 3 |
| -3 -2 |
Step 3: Find the determinant of the coefficient matrix:
To find the determinant, we use the formula:
Determinant = (ad - bc), where a, b, c, and d are the elements of the matrix.
In this case, the determinant is:
Determinant = (2 * -2) - (3 * -3)
Determinant = (-4) - (-9)
Determinant = -4 + 9
Determinant = 5
Step 4: Determine the number of solutions:
The number of solutions is determined by the value of the determinant.
If the determinant is non-zero (i.e., not equal to zero), there is exactly one solution.
If the determinant is zero, there can either be infinitely many solutions or no solutions.
In this case, since the determinant is 5 (non-zero), the system of equations has one solution.
Therefore, the correct answer is "one solution."
To find the number of solutions for the given system of equations, we can use the method of elimination or substitution.
Using the method of elimination, we can eliminate one variable by adding or subtracting the two equations. Let's multiply the first equation by 2 and the second equation by 3 to make the coefficients of x in both equations opposite:
2 * (2x + 3y) = 2 * 1
3 * (-3x - 2y) = 3 * (-1)
This simplifies to:
4x + 6y = 2
-9x - 6y = -3
Now, if we add the two equations together, the y variables will cancel out:
(4x + 6y) + (-9x - 6y) = 2 + (-3)
-5x + 0y = -1
-5x = -1
Dividing both sides of the equation by -5, we get:
x = 1/5
Now, substitute this value back into one of the original equations to solve for y. Let's use the first equation:
2(1/5) + 3y = 1
2/5 + 3y = 1
3y = 1 - 2/5
3y = 5/5 - 2/5
3y = 3/5
Dividing both sides of the equation by 3, we get:
y = 1/5
Therefore, the given system of equations has one solution, with x = 1/5 and y = 1/5.