1. **State the problem:** Solve the trigonometric equation $$7\cos(x) - 5\sin(x) = 5$$ for $x$. 2. **Use the formula:** We can express $a\cos(x) + b\sin(x)$ as $R\cos(x - \alpha)$ where $$R = \sqrt{a^2 + b^2}$$ and $$\tan(\alpha) = \frac{b}{a}$$. 3. **Calculate $R$ and $\alpha$:** $$R = \sqrt{7^2 + (-5)^2} = \sqrt{49 + 25} = \sqrt{74}$$ $$\tan(\alpha) = \frac{-5}{7}$$ 4. **Rewrite the equation:** $$7\cos(x) - 5\sin(x) = R\cos(x - \alpha) = 5$$ 5. **Divide both sides by $R$:** $$\frac{R\cos(x - \alpha)}{\cancel{R}} = \frac{5}{\cancel{R}} \Rightarrow \cos(x - \alpha) = \frac{5}{\sqrt{74}}$$ 6. **Solve for $x - \alpha$:** $$x - \alpha = \pm \arccos\left(\frac{5}{\sqrt{74}}\right) + 2k\pi, \quad k \in \mathbb{Z}$$ 7. **Find $\alpha$:** $$\alpha = \arctan\left(\frac{-5}{7}\right)$$ 8. **Final solution:** $$x = \alpha \pm \arccos\left(\frac{5}{\sqrt{74}}\right) + 2k\pi, \quad k \in \mathbb{Z}$$ This gives all solutions for $x$ in terms of $k$.