To solve the system of linear equations you provided, we need to express each equation in a standard form and then analyze them. Let's first rewrite the equations clearly.
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The first equation is: \[ 3y = \frac{3}{2}x + 6. \] We can rearrange this to: \[ 3y - \frac{3}{2}x - 6 = 0. \]
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The second equation is: \[ y - \frac{1}{2}y - \frac{1}{4}x = 3. \] We simplify this: \[ \frac{1}{2}y - \frac{1}{4}x = 3. \] Rearranging gives us: \[ \frac{1}{2}y = \frac{1}{4}x + 3. \] This can be rewritten as: \[ 1y - \frac{1}{2}x - 6 = 0. \]
Now, we compose a system of equations:
- \(3y - \frac{3}{2}x - 6 = 0\) (Let's call this equation 1)
- \(y - \frac{1}{2}x - 6 = 0\) (Let's call this equation 2)
To solve this system, we can manipulate these equations to analyze them further:
From equation 2: \[ y = \frac{1}{2}x + 6. \]
Now, we can substitute the value of \(y\) into equation 1: \[ 3\left(\frac{1}{2}x + 6\right) - \frac{3}{2}x - 6 = 0. \] Expanding this: \[ \frac{3}{2}x + 18 - \frac{3}{2}x - 6 = 0, \] which simplifies to: \[ 18 - 6 = 0 \rightarrow 12 = 0. \] This is a contradiction, indicating that the two equations do not intersect.
Since we arrive at a contradiction, the conclusion is that the system of equations has no solution. Therefore, the correct answer is "no solution."