Question

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

Explanation:

Step1: Simplify first equation

Divide both sides by 3 to isolate $y$.
$$y = \frac{1}{2}x + 2$$

Step2: Simplify second equation

Multiply by 4, rearrange to isolate $y$.
$$2y - x = 12 \implies y = \frac{1}{2}x + 6$$

Step3: Compare the two equations

Both have the same slope $\frac{1}{2}$ but different y-intercepts (2 vs. 6), so they are parallel lines that never intersect.

Answer:

no solution