Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.
−5x+3y=-9
−20x+11y=−48
To determine if the system of equations has no solutions, infinitely many solutions, or exactly one solution, we can use the method of elimination or substitution to solve the system of equations.
Let's use the method of elimination:
Multiply the first equation by 4 to make the coefficients of x in both equations the same:
-20x + 12y = -36
Now, we can subtract this new equation from the second equation:
(-20x + 11y) - (-20x + 12y) = -48 - (-36)
-20x + 11y + 20x - 12y = -48 + 36
-y = -12
Divide both sides of the equation by -1 to solve for y:
y = 12
Now that we have the value of y, we can substitute it back into one of the original equations to solve for x:
-5x + 3(12) = -9
-5x + 36 = -9
-5x = -45
x = 9
Therefore, the system of equations has exactly one solution: x = 9 and y = 12.