Final answer:
To solve the given system using elimination, we manipulate the equations by finding the least common multiple of the coefficients of y and multiply each equation by appropriate values to make the y coefficients opposites. Upon adding the resulting equations, we see that they represent the same line, indicating infinitely many solutions.
Step-by-step explanation:
To solve the system of equations using elimination, we need to manipulate one or both equations such that adding or subtracting one equation from the other eliminates one of the variables. You are given the equations 6x - 3y = 12 and -4x + 2y = -8.
- First, we observe that the coefficients of y are -3 and 2. The Least Common Multiple (LCM) of 3 and 2 is 6. We can multiply the first equation by 2 and the second equation by 3 to get the coefficients of y to be -6 and 6 respectively.
- Multiply the first equation: (6x - 3y) × 2 = 12 × 2 which yields 12x - 6y = 24.
- Multiply the second equation: (-4x + 2y) × 3 = -8 × 3 which yields -12x + 6y = -24.
- Now we can add the two new equations together to eliminate y: (12x - 6y) + (-12x + 6y) = 24 + (-24), resulting in 0x + 0y = 0. This indicates the equations are dependent and represent the same line.
Since the equations represent the same line, there is not a single solution, but rather infinitely many solutions (all points on the line that the equation represents).