How many solutions do the system of equations have -5x+y=-18 and 10x-2y=36
The system of equations has one solution.
To find the number of solutions in a system of equations, we can use the method of elimination or substitution. Let's use the method of elimination in this case.
Given the system of equations:
-5x + y = -18 ...(Equation 1)
10x - 2y = 36 ...(Equation 2)
Multiply Equation 1 by 2 to make the coefficients of y in both equations equal:
-10x + 2y = -36 ...(Equation 3)
Now, we can add Equation 2 and Equation 3:
(10x - 2y) + (-10x + 2y) = 36 + (-36)
0 = 0
The result is 0 = 0, which means the two equations are equivalent. This indicates that the system of equations has infinitely many solutions.
In conclusion, the given system of equations has infinitely many solutions.
To find the number of solutions for the given system of equations, we can use the method of elimination or substitution.
Let's use the method of elimination:
Step 1: Multiply the first equation by 2, and the second equation by 1 to make the coefficients of y opposite:
-10x + 2y = -36 (equation 1)
10x - 2y = 36 (equation 2)
Step 2: Add both equations together:
-10x + 2y + 10x - 2y = -36 + 36
0 = 0
Step 3: Simplify the equation:
0 = 0
The equation 0 = 0 is always true, which means that both equations represent the same line. Therefore, the system of equations has infinitely many solutions.
Explanation:
When both equations in a system of equations represent the same line, they are essentially describing the same relationship between the variables. This means that every point on the line satisfies both equations, resulting in an infinite number of solutions.