Question

what is the solution of this system of linear equations?
$3y = \frac{3}{2}x + 6$
$\frac{1}{2}y - \frac{1}{4}x = 3$
\\(\boldsymbol{\circ}\\) \\((3, 6)\\)
\\(\boldsymbol{\circ}\\) \\((2, 1)\\)
\\(\boldsymbol{\circ}\\) no solution
\\(\boldsymbol{\circ}\\) infinite number of solutions

Explanation:

Step1: Simplify the first equation

Divide the first equation \(3y = \frac{3}{2}x + 6\) by 3:
\(y=\frac{1}{2}x + 2\)

Step2: Simplify the second equation

Multiply the second equation \(\frac{1}{2}y-\frac{1}{4}x = 3\) by 4 to eliminate fractions:
\(2y - x=12\)
Then, solve for \(y\):
\(2y=x + 12\)
\(y=\frac{1}{2}x+6\)

Step3: Analyze the slopes and intercepts

The first equation \(y=\frac{1}{2}x + 2\) has a slope of \(\frac{1}{2}\) and a y - intercept of 2.
The second equation \(y=\frac{1}{2}x+6\) has a slope of \(\frac{1}{2}\) and a y - intercept of 6.
Since the two lines have the same slope (\(m_1 = m_2=\frac{1}{2}\)) but different y - intercepts (\(b_1 = 2
eq b_2 = 6\)), the lines are parallel and never intersect.

Answer:

no solution