Final answer:
The system of equations 6x−3y=15 and 8x−4y=20 is dependent, meaning the equations are multiples of each other and there are infinitely many solutions. The solution cannot be expressed as a single ordered pair, but any value of x can be used to solve for the corresponding y, and vice versa.
Step-by-step explanation:
To find the solution for the system of equations 6x−3y=15 and 8x−4y=20, we can start by simplifying the equations. Notice that both equations have similar terms for x and y. We can divide the second equation by 2 to get an equivalent equation that will help us compare it directly to the first one:
8x - 4y = 20 divided by 2 gives us 4x - 2y = 10.
Now, let's analyze the simplified system:
- 6x - 3y = 15 (Equation 1)
- 4x - 2y = 10 (Equation 2)
Notice that if we multiply Equation 2 by 1.5, the coefficients of y will match:
1.5(4x - 2y) = 1.5(10) gives us 6x - 3y = 15, which is the same as Equation 1. This means that the equations are not distinct but rather multiples of each other.
Since the equations are equivalent, there are infinitely many solutions that satisfy both equations. This implies that the system is dependent and the solutions can be described by setting x or y to a particular value and solving for the other variable.