How many solutions does the system have?
x = -4y + 4
2x + 8y = 8
A. one solution
B. two solutions
C. infinitely many solutions
D. no solution
The Correct Answer and Explanation is :
The correct answer is: C. infinitely many solutions
To determine the number of solutions for the given system of equations, we can start by rewriting the equations:
- ( x = -4y + 4 ) (Equation 1)
- ( 2x + 8y = 8 ) (Equation 2)
Step 1: Substitute Equation 1 into Equation 2
We can substitute the expression for ( x ) from Equation 1 into Equation 2:
[
2(-4y + 4) + 8y = 8
]
Step 2: Simplify the Equation
Now, distribute the 2 in the equation:
[
-8y + 8 + 8y = 8
]
Combine like terms:
[
0y + 8 = 8
]
This simplifies to:
[
8 = 8
]
Step 3: Analyze the Result
The equation ( 8 = 8 ) is always true and does not depend on ( y ). This means that for any value of ( y ), there is a corresponding value of ( x ) given by Equation 1. In other words, there are infinitely many combinations of ( x ) and ( y ) that satisfy both equations.
Conclusion
Since we found that the second equation is satisfied for all values of ( y ), the system of equations has infinitely many solutions. Each solution corresponds to a different value of ( y ) and a specific value of ( x ) calculated from Equation 1.
Thus, the correct answer is C. infinitely many solutions. This type of system is called dependent, as the equations represent the same line when graphed, confirming that they have the same set of solutions.