Answer
$$x=5$$ $$y=-2$$
Work Step by Step
Equation 1: $\frac{x-3}{2}=\frac{2-y}{4}$
Equation 2: $\frac{7-2x}{3}=\frac{y}{2}$
Simplify the equations by cross-multiplying the numerator of the left-hand side with the denominator of the right-hand side, and the numerator of the right-hand side with the denominator of the left-hand side.
Equation 1: $$\frac{x-3}{2}=\frac{2-y}{4}$$ $$(x-3)(4)=(2-y)(2)$$ $$4x-12=4-2y$$
We can call this equation 1'.
Equation 2: $$\frac{7-2x}{3}=\frac{y}{2}$$ $$(7-2x)(2)=3(y)$$ $$14-4x=3y$$
We can call this equation 2'.
Using equation 2, divide both sides by $3$:
$$14-4x=3y$$ $$\frac{14-4x}{3}=\frac{3y}{3}$$ $$\frac{14-4x}{3}=y$$
Substitute this equation to equation 1':
$$4x-12=4-2(\frac{14-4x}{3})$$ $$4x-12=4-(\frac{28-8x}{3})$$
Multiply the whole equation by $3$:
$$[4x-12=4-(\frac{28-8x}{3})]\cdot3$$ $$12x-36=12-(28-8x)$$ $$12x-36=12-28+8x$$ $$12x-36=-16+8x$$
Add $36$ to both sides:
$$12x-36=-16+8x$$ $$12x-36+36=-16+8x+36$$ $$12x=8x+20$$
Subtract $8x$ from both sides:
$$12x=8x+20$$ $$12x-8x=8x-8x+20$$ $$4x=20$$
Divide both sides by $4$:
$$\frac{4x}{4}=\frac{20}{4}$$ $$x=5$$
Substitute this value of $x$ to equation 1':
$$4x-12=4-2y$$ $$4(5)-12=4-2y$$ $$20-12=4-2y$$ $$8=4-2y$$
Subtract $4$ from both sides:
$$8-4=4-4-2y$$ $$4=-2y$$
Divide both sides by $-2$:
$$\frac{4}{-2}=\frac{-2y}{-2}$$ $$-2=y$$ $$y=-2$$
Check using equation 2':
$$14-4x=3y$$ $$14-4(5)=3(-2)$$ $$14-20=-6$$ $$-6=-6$$