solve the system by the method of your choice.identify systems with no solution and systems with infinitely many solutions using set notation to express their solution sets. 4x-3y=6 -12x+9y=-24
multiply the first by 3 -----> 12x - 9y = 18
add it to the 2nd : ---------> -12x + 9y = -24
add them to get: 0 = -6
which is a false statement
Thus there is no solution to this system
(notice they are two distinct parallel lines, which can never intersect)
State the result in the notation that you learned.
To solve the given system of equations, we will use the method of elimination.
Step 1: Multiply the first equation by 12 and the second equation by 4 to make the coefficients of x terms in both equations equal.
12(4x - 3y) = 12(6) ------> equation (1)
4(-12x + 9y) = 4(-24) ------> equation (2)
Simplifying the equations, we get:
48x - 36y = 72 ------> equation (3)
-48x + 36y = -96 ------> equation (4)
Step 2: Add equations (3) and (4) to eliminate the x variable.
(48x - 36y) + (-48x + 36y) = 72 + (-96)
By simplifying, we obtain:
0 = -24
Step 3: Analyze the obtained result.
The equation 0 = -24 is inconsistent and cannot be satisfied. It implies that there is no solution to the system of equations. Hence, the given system has no solution.
Now, let's express this result using set notation:
As the system has no solution, we represent it as an empty or null set: ∅.
Answer: The given system of equations has no solution, and its solution set is empty (∅).