Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
2, x, plus, y, equals, minus, 3
2x+y=
−3
minus, 2, x, minus, y, equals, 5
−2x−y=
5
To determine the number of solutions for the given system of equations, we can solve it using elimination or substitution.
Using elimination:
To eliminate the variable "y," we can multiply the second equation by -1 and add it to the first equation:
2x + y = -3
-(-2x - y = 5)
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4x = 2
Dividing both sides of the equation by 4, we get:
x = 1/2
Now, we can substitute the value of x into one of the equations to solve for y:
2(1/2) + y = -3
1 + y = -3
y = -4
Therefore, the system of equations has exactly one solution: x = 1/2 and y = -4.
Thus, the correct answer is: exactly one solution.
To determine if the given system of equations has no solutions, infinitely many solutions, or exactly one solution, we can solve the system using either substitution or elimination method.
The given system of equations is:
1) 2x + y = -3
2) -2x - y = 5
Let's solve the system using the elimination method:
To eliminate the variable "y", we will multiply equation 1) by -1 and then add it to equation 2):
-1 * (2x + y) = -1 * (-3)
-2x - y = 3
Adding this to equation 2):
-2x - y + (-2x - y) = 5 + 3
-4x = 8
Dividing both sides of the equation by -4:
-4x / -4 = 8 / -4
x = -2
Now, substitute the value of x = -2 into equation 1):
2(-2) + y = -3
-4 + y = -3
y = -3 + 4
y = 1
Therefore, the solution to the given system of equations is x = -2 and y = 1.
Since there is a unique solution, the given system has exactly one solution.
To determine if the given system of equations has no solutions, infinitely many solutions, or exactly one solution, we can use the method of elimination or substitution. Let's use the method of elimination here.
Step 1: Write down the equations using the given information:
Equation 1: 2x + y = -3
Equation 2: -2x - y = 5
Step 2: Multiply equation 1 by 2 to create opposite coefficients for the x terms:
2 * (2x + y) = 2 * (-3)
4x + 2y = -6
Now, we have the following equations:
Equation 1: 4x + 2y = -6
Equation 2: -2x - y = 5
Step 3: Add equation 1 and equation 2 together to eliminate the y term:
(4x + 2y) + (-2x -y) = -6 + 5
4x + 2y - 2x - y = -1
2x + y = -1
Now we have a new equation:
Equation 3: 2x + y = -1
Step 4: Compare Equation 3 with Equation 1:
Equation 3: 2x + y = -1
Equation 1: 2x + y = -3
These equations have the same coefficients for x and y terms and equal constants, but different results. Therefore, the system of equations has no solutions.
So, the answer is: The given system of equations has no solutions.