What solution(s) does the system of equations have?
-2x + 5y = 10
-4x + 10y = 20
A: There are infinitely many solutions.
B: The only solution is (5,4)
C: The only solution is (10,6)
D: There are no solutions.
if you multiply the 1st equation by two, you get the 2nd equation
... the two equations are basically the same
what does that mean for solutions?
anything that works for the 1st, will also work for the 2nd
it asked if it had man solutions or one solution or no solution
To find the solution(s) of the system of equations, let's solve it step-by-step.
Step 1: We have the following system of equations:
-2x + 5y = 10 ... (Equation 1)
-4x + 10y = 20 ... (Equation 2)
Step 2: Multiply Equation 1 by -2 to eliminate x:
-2(-2x + 5y) = -2(10)
4x - 10y = -20
Step 3: Now, let's add Equation 2 to the result of Step 2:
(4x - 10y) + (-4x + 10y) = -20 + 20
4x - 10y - 4x + 10y = 0
0 = 0
Step 4: The result in Step 3 indicates that both equations are the same, or they represent the same line. This means that the system has infinitely many solutions.
Therefore, the correct answer is:
A: There are infinitely many solutions.
To determine the solution(s) to the system of equations, we can solve it using the method of elimination or substitution. Let's use the method of elimination to solve this system.
First, multiply both sides of the first equation by 2 to eliminate the x term:
-4x + 10y = 20
Now we have the system of equations:
-4x + 10y = 20
-4x + 10y = 20
We can see that both equations are identical. This means that the two equations represent the same line. Therefore, the system of equations has infinitely many solutions.
So, the correct answer is A: There are infinitely many solutions.