Which system has an infinite number of solutions?
A.{x+2=y 4=2y-x
B.{2y+6=4x -3=y-2x
C.{y+3=2x 4x=2y-3
D.{y=2x-5 -2=y-2x
Is it B
Yes it is. :)
To determine whether a system of equations has an infinite number of solutions, you need to check if the two equations are equivalent or if one equation can be derived from the other.
Let's analyze the given options:
A. {x+2=y 4=2y-x
The first equation (x+2=y) cannot be derived from the second equation (4=2y-x), and the second equation cannot be derived from the first equation. Thus, this system does not have an infinite number of solutions.
B. {2y+6=4x -3=y-2x
By rearranging the second equation (-3=y-2x), we can rewrite it as y=2x-3. Now, if we compare the first equation (2y+6=4x) with y=2x-3, we can see that they are equivalent. Therefore, this system has an infinite number of solutions.
C. {y+3=2x 4x=2y-3
The two equations in this system cannot be rearranged to be equivalent or derived from one another. Thus, this system does not have an infinite number of solutions.
D. {y=2x-5 -2=y-2x
The first equation (y=2x-5) cannot be derived from the second equation (-2=y-2x), and the second equation cannot be derived from the first equation. Hence, this system does not have an infinite number of solutions.
Therefore, the correct option with a system of equations that has an infinite number of solutions is B. {2y+6=4x -3=y-2x.
To determine which system has an infinite number of solutions, we need to solve each system of equations and see if they have a unique solution or not.
Let's solve each system:
Option A:
x + 2 = y
4 = 2y - x
To solve this system, we can start by isolating one variable and substituting it into the other equation. Let's isolate x in the first equation:
x = y - 2
Now substitute this value of x into the second equation:
4 = 2y - (y - 2)
4 = 2y - y + 2
4 = y + 2
y = 2
Substituting the value of y back into the equation x = y - 2:
x = 2 - 2
x = 0
So in option A, the solution is x = 0 and y = 2, meaning it has a unique solution.
Option B:
2y + 6 = 4x
-3 = y - 2x
Similarly, let's isolate one variable and substitute it into the other equation. Let's isolate y in the second equation:
y = 2x - 3
Now substitute this value of y into the first equation:
2(2x - 3) + 6 = 4x
4x - 6 + 6 = 4x
4x = 4x
In this case, the variable x cancels out, meaning any value of x would satisfy these equations. Thus, this system has an infinite number of solutions.
Therefore, the correct answer is option B.