1. **Determine whether each equation is linear in the given variables.** Linear equations in variables $x_1, x_2, x_3$ or $x, y$ have the form $$a_1 x_1 + a_2 x_2 + a_3 x_3 = b$$ or $$a x + b y = c$$ where $a_i, a, b, c$ are constants and variables appear only to the first power, not multiplied together or inside functions like roots, powers, or trigonometric functions. --- **Group 1: Variables $x_1, x_2, x_3$** a. $x_1 + 5x_2 - \sqrt{2} x_3 = 1$ - All variables appear to the first power and are not multiplied together. - **Linear**. b. $x_1 + 3x_2 + x_1 x_3 = 2$ - Term $x_1 x_3$ is a product of variables. - **Not linear**. c. $x_1 = -7x_2 + 3x_3$ - Variables appear to the first power, no products. - **Linear**. d. $x_1^{-2} + x_2 + 8x_3 = 5$ - $x_1^{-2}$ is $x_1$ to the power $-2$, not linear. - **Not linear**. e. $x_1^{3/5} - 2x_2 + x_3 = 4$ - $x_1^{3/5}$ is a fractional power, not linear. - **Not linear**. f. $\pi x_1 - \sqrt{2} x_2 = 7^{1/3}$ - Variables appear linearly multiplied by constants. - **Linear**. --- **Group 2: Variables $x, y$** a. $2^{1/3} x + \sqrt{3} y = 1$ - Variables appear linearly. - **Linear**. b. $2 x^{1/3} + 3 \sqrt{y} = 1$ - Variables inside roots and fractional powers. - **Not linear**. c. $\cos(\pi/7) x - 4 y = \log 3$ - Variables multiplied by constants, no nonlinear functions of variables. - **Linear**. d. $\frac{\pi}{7} \cos x - 4 y = 0$ - $\cos x$ is a nonlinear function of $x$. - **Not linear**. e. $x y = 1$ - Product of variables. - **Not linear**. f. $y + 7 = x$ - Variables appear linearly. - **Linear**. --- **Final answers:** Group 1 linear: a, c, f Group 1 not linear: b, d, e Group 2 linear: a, c, f Group 2 not linear: b, d, e