How many solution(s) does the system of equations have? −5x+y=−18 10x−2y=36 (1 point) Responses none none two two infinite infinite one
To determine the number of solutions for the system of equations, we can solve the equations and observe the results.
The given system of equations is:
-5x + y = -18 ......(1)
10x - 2y = 36 ......(2)
Let's solve the system using the method of elimination:
Multiply equation (1) by 2 to make the coefficients of y the same in both equations:
-10x + 2y = -36 ......(3)
Add equations (2) and (3) together:
(10x - 2y) + (-10x + 2y) = 36 + (-36)
0 = 0
The result is 0 = 0, which means the two equations are equivalent. This indicates that they represent the same line. In other words, the lines are coincident or overlapping.
Since the lines coincide, the system of equations has infinite solutions.
So, the answer is: infinite (infinite solutions).
To determine the number of solutions for a system of equations, we need to solve the equations simultaneously and see how many solutions we obtain.
Let's solve the given system of equations:
Equation 1: -5x + y = -18
Equation 2: 10x - 2y = 36
We can solve this system by using the method of substitution or elimination.
Method 1: Substitution
Step 1: Solve Equation 1 for y:
-5x + y = -18
y = -18 + 5x
Step 2: Substitute the value of y into Equation 2 and solve for x:
10x - 2(-18 + 5x) = 36
10x + 36 - 10x = 36
36 = 36
Since the x variable cancels out, we can conclude that there is no specific value of x. However, since y is dependent on x, there is an infinite number of possible values for y that satisfy the equation.
Therefore, this system of equations has an infinite number of solutions.
Response: infinite