How many solutions does the following system have? {2x+3y=1−3x−2y=−1 (1 point) Responses Two solutions Two solutions Infinitely many solutions Infinitely many solutions No solutions No solutions One solution
This system of equations has one solution.
Let's solve the system of equations step by step to determine how many solutions it has.
First, let's rewrite the system:
2x + 3y = 1 (Equation 1)
-3x - 2y = -1 (Equation 2)
To eliminate the variable x, let's multiply Equation 2 by 2:
-6x - 4y = -2 (Equation 3)
Now, add Equation 1 and Equation 3 together:
(2x + 3y) + (-6x - 4y) = (1) + (-2)
Simplifying the left side of the equation:
-4x - y = -1 (Equation 4)
Now, let's rewrite Equation 4 to solve for y:
y = -4x - 1 (Equation 5)
Substituting Equation 5 into Equation 1:
2x + 3(-4x - 1) = 1
Simplifying:
2x - 12x - 3 = 1
-10x = 4
x = -4/10
x = -2/5
Now, let's substitute the value of x into Equation 5 to find the value of y:
y = -4(-2/5) - 1
y = 8/5 - 1
y = 8/5 - 5/5
y = 3/5
Therefore, the solution to the system of equations is x = -2/5 and y = 3/5.
Since we have found unique values for both x and y, the system has one unique solution.
The correct answer is: One solution.
To determine the number of solutions for a system of linear equations, we can use either the substitution method or the elimination method. Let's solve this system using the elimination method:
The given system of equations is:
2x + 3y = 1 (Equation 1)
-3x - 2y = -1 (Equation 2)
To eliminate one of the variables, we can multiply Equation 1 by 3 and Equation 2 by 2:
6x + 9y = 3 (Equation 3)
-6x - 4y = -2 (Equation 4)
Now, add Equation 3 and Equation 4 together:
(6x + 9y) + (-6x - 4y) = 3 + (-2)
6x - 6x + 9y - 4y = 1
5y = 1
Divide both sides of the equation by 5:
y = 1/5
Now, we can substitute this value of y back into either Equation 1 or Equation 2. Let's use Equation 1:
2x + 3(1/5) = 1
2x + 3/5 = 1
2x = 1 - 3/5
2x = 2/5
Divide both sides of this equation by 2:
x = 1/5
So, we have found the values of x and y, which are x = 1/5 and y = 1/5. Therefore, this system of equations has one solution.
The correct answer is: One solution.