How many solutions does the system of equations have?
2x = -10y + 6 and x + 5y =3
A: one
B: two
C: infinitely many
D: none
_________________
I think it is either A or D, A because i graphed it and i could only graph 2x = -10y + 6, and maybe D because i couldn't get a complete solution
don't guess. if you rearrange the equations, you get
2x+10y = 6
x+5y = 3
Those are the same line (divide the top by 2 and you get the bottom)
So, C
To determine the number of solutions for the system of equations, we can use the method of elimination or substitution.
First, let's solve the second equation for x:
x + 5y = 3
x = 3 - 5y
Now, substitute this expression for x into the first equation:
2x = -10y + 6
2(3 - 5y) = -10y + 6
6 - 10y = -10y + 6
Notice that the variable -10y cancels out on both sides of the equation. This tells us that the value of y does not affect the equation.
As a result, we are left with 6 = 6, which is a true statement.
Since the two equations are not independent and lead to a true statement, the system of equations has infinitely many solutions (C).
To determine the number of solutions for the given system of equations, let's solve it algebraically.
The given system of equations is:
1) 2x = -10y + 6
2) x + 5y = 3
To solve this system, we can use the method of substitution or elimination. Let's choose the method of substitution.
From equation 1), we can isolate x:
2x = -10y + 6
Divide both sides by 2:
x = -5y + 3
Now, substitute this value of x into equation 2):
-5y + 3 + 5y = 3
Combine like terms:
-5y + 5y + 3 = 3
3 = 3
This equation simplifies to 3 = 3, which means the equation is true.
Since the equation 3 = 3 is true, it shows that the two equations in the system are equivalent, representing the same line.
Hence, the system has infinitely many solutions.
Therefore, the correct answer is C: infinitely many.