Solve the following system. x 2 – y = 3 x – y = -3 The solution set
The Correct Answer and Explanation is:
We are given the system of equations:
- ( x^2 – y = 3 )
- ( x – y = -3 )
Step 1: Solve for ( y ) in the second equation
From the second equation, ( x – y = -3 ), we can isolate ( y ) as follows:
[
x – y = -3
]
[
y = x + 3
]
Step 2: Substitute the expression for ( y ) into the first equation
Now that we have ( y = x + 3 ), substitute this into the first equation ( x^2 – y = 3 ):
[
x^2 – (x + 3) = 3
]
Simplify the equation:
[
x^2 – x – 3 = 3
]
Step 3: Solve the resulting quadratic equation
To solve for ( x ), first move all terms to one side:
[
x^2 – x – 3 – 3 = 0
]
[
x^2 – x – 6 = 0
]
Now factor the quadratic equation:
[
x^2 – x – 6 = (x – 3)(x + 2) = 0
]
Set each factor equal to zero:
[
x – 3 = 0 \quad \text{or} \quad x + 2 = 0
]
Solving these gives:
[
x = 3 \quad \text{or} \quad x = -2
]
Step 4: Find the corresponding values of ( y )
We now substitute these values of ( x ) into the equation ( y = x + 3 ) to find the corresponding values of ( y ):
- If ( x = 3 ), then ( y = 3 + 3 = 6 ).
- If ( x = -2 ), then ( y = -2 + 3 = 1 ).
Step 5: Conclusion
The solution set is ( (3, 6) ) and ( (-2, 1) ).
Thus, the solutions to the system are ( \boxed{(3, 6) \text{ and } (-2, 1)} ).
Explanation
We started by isolating one variable, ( y ), from the second equation. We substituted this expression into the first equation, resulting in a quadratic equation. After solving the quadratic, we found two possible values for ( x ) and then substituted those into the equation for ( y ) to find the corresponding values. This approach ensures that both equations are satisfied by the solutions we found.