How many solution(s) does the system of equations have?
−5x+y=−18
10x−2y=36 (1 point)
Responses
infinite
infinite
one
one
two
two
none
one
To find out how many solutions the system of equations has, we can solve the equations using a method such as substitution or elimination. Let's use the elimination method to solve this system:
The given equations are:
-5x + y = -18 (Equation 1)
10x - 2y = 36 (Equation 2)
To eliminate the variable "y", let's multiply Equation 1 by 2:
-10x + 2y = -36 (Equation 3)
Now, we can add Equation 2 and Equation 3 together:
(10x - 2y) + (-10x + 2y) = 36 + (-36)
0 = 0
Since the equation 0 = 0 is true, it means that the system of equations is dependent or consistent. This means that the two equations represent the same line, and there are infinitely many solutions to this system.
Therefore, the correct answer is: infinite.
To determine the number of solutions for a system of equations, we can use the method of elimination or substitution. Let's solve the system of equations to find the number of solutions.
First, let's rewrite the equations in standard form:
Equation 1: -5x + y = -18
Equation 2: 10x - 2y = 36
Now, to eliminate one variable, we can multiply Equation 1 by 2:
2(-5x + y) = 2(-18)
-10x + 2y = -36
Now, subtract Equation 2 from the new Equation 1:
(-10x + 2y) - (10x - 2y) = (-36) - (36)
-10x + 10x + 2y - (-2y) = -72
0 + 4y = -72
4y = -72
Divide both sides by 4 to solve for y:
y = -72/4
y = -18
Now, substitute the value of y back into either of the original equations. Let's use Equation 1:
-5x + (-18) = -18
-5x - 18 = -18
Add 18 to both sides:
-5x = 0
Divide both sides by -5:
x = 0
So, we have found the solution for x and y.
The system of equations has only one solution, which is x = 0 and y = -18. Therefore, the answer is "one".