How many solutions does the following system have?
2x+3y=1
-3x-2y=-1
A. Infinitely many solutions
B. two solutions
C. one solution
D. no solutions
To solve this system, we can multiply the first equation by 2 and the second equation by 3 in order to eliminate x.
Multiplying the first equation by 2 gives:
4x + 6y = 2.
Multiplying the second equation by 3 gives:
-9x - 6y = -3.
Adding the two new equations together eliminates y:
4x - 9x + 6y - 6y = 2 - 3.
-5x = -1.
Dividing both sides of the equation by -5 gives:
x = 1/5.
Substituting this value for x in the first equation:
2(1/5) + 3y = 1.
2/5 + 3y = 1.
Subtracting 2/5 from both sides of the equation gives:
3y = 1 - 2/5.
3y = 5/5 - 2/5.
3y = 3/5.
Dividing both sides of the equation by 3 gives:
y = 1/5.
So, the system has only one solution, which is x = 1/5 and y = 1/5.
Therefore, the correct answer is C. one solution.
To determine the number of solutions for a system of equations, we can use the method of substitution or elimination. Let's use the method of elimination to find the solution to this system.
The given system of equations is:
2x + 3y = 1 ----(1)
-3x - 2y = -1 ----(2)
To eliminate one of the variables, we can multiply both sides of equation (2) by 2 and equation (1) by 3. This will allow us to cancel out the y variable.
Multiplying equation (2) by 2, we have:
2*(-3x - 2y) = 2*(-1)
-6x - 4y = -2 ----(3)
Multiplying equation (1) by 3, we have:
3*(2x + 3y) = 3*(1)
6x + 9y = 3 ----(4)
Now, let's add equation (3) and equation (4) to eliminate the x variable:
(-6x - 4y) + (6x + 9y) = -2 + 3
-6x + 6x - 4y + 9y = 1
5y = 1
Simplifying, we find:
5y = 1
Now, divide both sides of the equation by 5 to solve for y:
y = 1/5
Substitute this value of y back into either equation (1) or (2) to find the value of x. Let's use equation (1):
2x + 3*(1/5) = 1
2x + 3/5 = 1
2x = 1 - 3/5
2x = 5/5 - 3/5
2x = 2/5
x = (2/5) / 2
x = (2/5) * (1/2)
x = 1/5
So, the solution to this system of equations is x = 1/5 and y = 1/5. Therefore, the system has only one solution.
The answer is C. one solution.
To determine the number of solutions for the given system of equations, we can use the method of elimination or substitution.
Let's solve the system using the elimination method:
Multiply the first equation by 3, and the second equation by 2, to make the coefficients of y in both equations equal:
3(2x + 3y) = 3(1) --> 6x + 9y = 3
2(-3x - 2y) = 2(-1) --> -6x - 4y = -2
Now, add the two equations together to eliminate the x variable:
(6x + 9y) + (-6x - 4y) = 3 + (-2)
6x - 6x + 9y - 4y = 3 - 2
5y = 1
Divide both sides of the equation by 5:
5y/5 = 1/5
y = 1/5
Now, substitute the value of y back into either of the original equations to solve for x. Let's use the first equation:
2x + 3(1/5) = 1
2x + 3/5 = 1
2x = 1 - 3/5
2x = 5/5 - 3/5
2x = 2/5
x = 1/5
Therefore, the system has only one solution: x = 1/5, y = 1/5.
The answer is C. one solution.