How many solutions does this system of equations have ??
2x= 10y + 6 and x+ 5y = 3
does it have infinitey many??
nvm 1 .
Can you give me the answers to the test lol
To determine the number of solutions, we need to solve the system of equations.
Let's solve the system step-by-step using the method of substitution.
From the second equation:
x + 5y = 3
We can rewrite it as:
x = 3 - 5y
Now substitute x in the first equation with the value from the second equation:
2(3 - 5y) = 10y + 6
Simplify the equation:
6 - 10y = 10y + 6
Combine like terms:
-10y - 10y = 6 - 6
Simplify further:
-20y = 0
Divide both sides of the equation by -20 to isolate y:
y = 0
Now substitute y back into one of the original equations, such as the second equation:
x + 5(0) = 3
x = 3
So the solution to the system of equations is x = 3 and y = 0.
Thus, the system has a unique solution, and it does not have infinitely many solutions.
To determine the number of solutions for the system of equations, let's solve it using the method of elimination or substitution.
Start with the given system of equations:
Equation 1: 2x = 10y + 6
Equation 2: x + 5y = 3
Method 1: Elimination
To eliminate one variable, we can multiply Equation 2 by 2 to match the coefficients of x in both equations:
2(x + 5y) = 2(3)
2x + 10y = 6
Now, we can subtract Equation 1 from this new equation:
(2x + 10y) - (2x) = 6 - (10y + 6)
0 = -4y - 6
Simplifying further, we can divide both sides by -2:
0 = 2y + 3
Since the equation 0 = 2y + 3 is inconsistent (0 cannot equal any value of 2y + 3), this suggests that the system of equations has NO solutions. Therefore, it does not have infinitely many solutions.
Method 2: Substitution
Solve Equation 2 for x:
x = 3 - 5y
Now substitute this expression for x in Equation 1:
2(3 - 5y) = 10y + 6
Simplify:
6 - 10y = 10y + 6
Rearrange the equation:
6 - 6 = 10y + 10y
0 = 20y
Divide by 20:
0 = y
Now substitute this value for y back into Equation 2:
x + 5(0) = 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = 0.
In conclusion, the system of equations has a unique solution (x = 3, y = 0).