How many solution(s) does the system of equations have?−5x+y=−18 10x−2y=36
To find the number of solutions a system of equations has, we can use the technique of graphing or solving algebraically.
First, let's solve this system of equations algebraically.
We can solve the first equation for y:
-5x + y = -18
y = 5x - 18
Now substitute this expression for y into the second equation:
10x - 2(5x - 18) = 36
10x - 10x + 36 = 36
0 + 36 = 36
The result is a true statement.
This means that the two equations are equivalent and represent the same line on the coordinate plane. Therefore, there are infinitely many solutions to this system of equations, meaning any point on the line given by y = 5x - 18 would satisfy both equations.
To determine the number of solutions for a system of equations, we can use the method of elimination or substitution. Let's use elimination to solve this system.
We have the following system of equations:
-5x + y = -18 ----(1)
10x - 2y = 36 ----(2)
To eliminate one of the variables, we can multiply equation (1) by 2 and equation (2) by 1, so that the coefficients of the y terms in both equations become opposites:
-10x + 2y = -36 ----(3)
10x - 2y = 36 ----(4)
Now we can add equations (3) and (4) together to eliminate the y variable:
(-10x + 2y) + (10x - 2y) = -36 + 36
The -10x and 10x terms cancel each other out, and the 2y and -2y terms also cancel each other out. On the right side, -36 + 36 equals 0, so the equation becomes:
0 = 0
This equation is always true, regardless of the values of x and y. This suggests that the two original equations are dependent, and there are infinitely many solutions to this system.
In conclusion, the given system of equations has infinitely many solutions.
To find the number of solutions for this system of equations, we can use the concept of elimination. Let's start by multiplying the first equation by 2 to make the coefficients of y the same:
-5x + y = -18 (equation 1)
10x - 2y = 36 (equation 2)
Multiplying the first equation by 2 gives:
-10x + 2y = -36 (equation 3)
Now, if we add equation 2 and equation 3, the x terms will cancel out:
(10x - 2y) + (-10x + 2y) = 36 - 36
0 = 0
This means that equation 2 and equation 3 are dependent and do not provide any new information. Therefore, we are left with:
-5x + y = -18 (equation 1)
0 = 0 (equation 4)
Equation 4 represents the statement "0 is equal to 0," which is always true. This means that equation 4 does not impose any constraints on the variables x and y.
Since equation 1 has two variables and one equation, there are infinitely many solutions for this system of equations.