To determine which equations can be used to solve for the value of [tex]\( z \)[/tex] using the Law of Sines, we need to evaluate each option through this formula:
[tex]\[ \frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c} \][/tex]
Let's analyze the given options:
1. [tex]\( \frac{\operatorname{mose}}{2.5} = \frac{3.95}{2} \)[/tex]
- This equation does not involve sine values or angles, and thus it does not follow the Law of Sines. Therefore, it cannot be used to solve for [tex]\( z \)[/tex].
2. [tex]\( \frac{\sin \left(51^{\circ}\right)}{2.5} = \frac{\sin \left(53^{\circ}\right)}{z} \)[/tex]
- This option matches the form required by the Law of Sines as it involves the sine of angles and the corresponding opposite sides. Thus, it can be used to solve for [tex]\( z \)[/tex].
3. [tex]\( \frac{\sin \left(76^{\circ}\right)}{2.5} = \frac{\sin \left(51^{\circ}\right)}{2} \)[/tex]
- This equation does not include [tex]\( z \)[/tex], so it cannot be used to solve for [tex]\( z \)[/tex].
4. [tex]\( \frac{\sin \left(75^{\circ}\right)}{2.6} = \frac{\sin \left(53^{\circ}\right)}{z} \)[/tex]
- This option also matches the form required by the Law of Sines as it involves the sine of angles and the corresponding opposite sides. Hence, it can be used to solve for [tex]\( z \)[/tex].
Therefore, the correct equations that can be used to solve for the value of [tex]\( z \)[/tex] are:
[tex]\[ \boxed{2 \text{ and } 4} \][/tex]
