The equations you've provided are:
- \( 3x + 2y = 12 \)
- \( 6x + 4y = 24 \)
Let's start by noticing the second equation can be simplified. Divide the entire equation by 2:
\[ \frac{6x}{2} + \frac{4y}{2} = \frac{24}{2} \]
This simplifies to:
\[ 3x + 2y = 12 \]
This means that both equations are actually the same. They represent the same line in the coordinate plane. Therefore, there are infinitely many solutions that lie along this line.
To express the relationship between \(x\) and \(y\), we can rearrange the first equation:
- \( 3x + 2y = 12 \)
Subtract \(3x\) from both sides:
\[ 2y = 12 - 3x \]
Now, divide by 2:
\[ y = 6 - \frac{3}{2}x \]
This is the equation of the line in slope-intercept form (y = mx + b), where the slope \(m = -\frac{3}{2}\) and the y-intercept \(b = 6\).
Thus, the solution set consists of all the points \((x, y)\) that satisfy this equation.