To solve the system of equations 3x + 2y = 6 and 6x + 4y = 12, we can use the method of substitution or elimination. However, we will start by examining the second equation.
The second equation 6x + 4y = 12 is actually a multiple of the first. If we divide the entire equation by 2, we get:
3x + 2y = 6
This means both equations represent the same line, which indicates that the system has infinitely many solutions. Every point (x, y) that lies on the line represented by 3x + 2y = 6 is a solution.
To find a specific solution, we can solve for y in terms of x using the first equation:
From 3x + 2y = 6, we isolate y:
2y = 6 – 3x
y = 3 – (3/2)x
This equation represents all the y-values corresponding to any x-value. For instance, if we let x = 0, we find:
y = 3 – (3/2)(0) = 3
So, one solution is (0, 3). Similarly, if we let x = 2, we get:
y = 3 – (3/2)(2) = 3 – 3 = 0
Thus, another solution is (2, 0). These examples illustrate that there are infinitely many solutions to the equations, all lying on the line defined by the first equation.