Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
−2x+3y=3
−4x+7y=9
To determine the number of solutions, we can use the method of solving the system of equations, such as substitution or elimination.
We will use the method of elimination. To do so, let's multiply the first equation by -2 to eliminate the x variable:
-2(-2x+3y) = -2(3)
4x-6y = -6
Now, we can write the updated system of equations:
4x-6y = -6
-4x+7y = 9
Adding the two equations, we get:
4x-6y + (-4x+7y) = -6 + 9
x + y = 3
Now we have the equation x + y = 3. Let's call this equation (1) and the equation -4x+7y = 9 equation (2).
From equation (1) we can solve for x in terms of y:
x = 3 - y
Now we substitute this expression for x in equation (2):
-4(3 - y) + 7y = 9
-12 + 4y + 7y = 9
11y - 12 = 9
11y = 9 + 12
11y = 21
y = 21/11
y = 3/2
Now we substitute the value of y = 3/2 back into equation (1):
x + (3/2) = 3
x = 3 - (3/2)
x = 3/2
Hence, the solution to the system of equations is x = 3/2 and y = 3/2.
Therefore, the system of equations has exactly one solution.