Question

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

Explanation:

Step1: Simplify first equation

Divide by 3:
$$y = \frac{1}{2}x + 2$$

Step2: Simplify second equation

Multiply by 4:
$$2y - x = 12$$
Rearrange to solve for $y$:
$$2y = x + 12$$
$$y = \frac{1}{2}x + 6$$

Step3: Compare the two lines

Check slopes and intercepts:
Both have slope $\frac{1}{2}$, but y-intercepts are 2 and 6 (different). Parallel lines never intersect.

Answer:

no solution