Question

Please Help Find the value of x

Answer 1

Let $\varphi$ be the measure of the angle labeled in the attached image below. By the law of sines,

$\dfrac{\sin\theta}{10x+5}=\dfrac{\sin(\pi-\varphi)}{78}=\dfrac{\sin\varphi}{78}\implies\sin\varphi=\dfrac{78\sin\theta}{10x+5}$

and

$\dfrac{\sin\theta}{6x-1}=\dfrac{\sin\varphi}{42}\implies\sin\varphi=\dfrac{42\sin\theta}{6x-1}$

Then

$\dfrac{78\sin\theta}{10x+5}=\dfrac{42\sin\theta}{6x-1}\implies78(6x-1)=42(10x+5)$

From here we can solve for $x$:

$468x-78=420x+210\implies48x=288\implies\boxed{x=6}$.