To solve the system of equations 3x + 2y = 6 and 6x + 4y = 12, we can use either the substitution method or the elimination method. Here, I’ll demonstrate the elimination method.
First, let’s write down the equations:
- Equation 1: 3x + 2y = 6
- Equation 2: 6x + 4y = 12
Notice that Equation 2 can be simplified. If we divide the entire equation by 2, we get:
3x + 2y = 6
This is actually the same as Equation 1. This indicates that both equations represent the same line in a two-dimensional space.
Since both equations are equivalent, there are infinitely many solutions along this line. In other words, the system is dependent.
To express the solutions, we can solve Equation 1 for y:
1. Start with the original equation: 3x + 2y = 6
2. Rearranging this gives us: 2y = 6 – 3x
3. Dividing both sides by 2 provides: y = 3 – (3/2)x
This means that for any value of x, we can find the corresponding value of y. For example:
- If x = 0, then y = 3
- If x = 2, then y = 0
- If x = 4, then y = -3
Thus, the solutions to the system of equations can be expressed as (x, y) = (x, 3 – (3/2)x) for any real number x.
In conclusion, the simultaneous solution of the equations indicates an infinite number of coordinate pairs (x, y) that satisfy both equations on the line described by 3x + 2y = 6.