Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.
x + y = -8
x - y = 14
A. {( 3, -11)}
B. {( 3, 11)}
C. {(x, y) | x + y = -8}
D. ∅
B?
x + y = -8
x - y = 14
x = -y - 8
-y - 8 - y = 14
-2y = 22
y = -11
x - 11 = -8
x = 3
To solve the given system of equations:
x + y = -8
x - y = 14
We can use the method of elimination. By adding the two equations together, we can eliminate the variable y and solve for x:
(x + y) + (x - y) = -8 + 14
2x = 6
x = 3
Now that we have the value of x, we can substitute it back into one of the original equations to solve for y:
x + y = -8
3 + y = -8
y = -8 - 3
y = -11
Therefore, the solution to the system of equations is (x, y) = (3, -11).
The correct answer is A. {( 3, -11)}
To solve the given system of equations:
1. Start by writing down the equations in the system:
Equation 1: x + y = -8
Equation 2: x - y = 14
2. Choose a method to solve the system. In this case, we will use the method of elimination. To eliminate the variable "y," we can add the two equations together.
(x + y) + (x - y) = -8 + 14
2x = 6
x = 6/2
x = 3
3. Substitute the value of x back into one of the original equations to solve for y.
x + y = -8
3 + y = -8
y = -8 - 3
y = -11
4. Check the solution by substituting the values of x and y into the second equation.
x - y = 14
3 - (-11) = 14
3 + 11 = 14
14 = 14
The solution to the system is x = 3 and y = -11, which matches option A: {( 3, -11)}. Therefore, option A is the correct answer.
Additionally, there is no need to consider options C and D since they are not valid solutions for this system. Option C represents the equation of one of the original equations in set notation, but it does not express the solution set itself. Option D, ∅ (an empty set), represents a system with no solutions, but that is not the case here. So, the correct answer is definitely option A.