Write the following sentences as algebraic equations and solve for the variable: a) Half a number, x, plus 17 is equal to 97. b) Double a number, y, plus 4 is equal to -4. c) Four times a number, z, i

"Half a number, $x$, plus 17 is equal to 97" translates to:

$$\frac{x}{2} + 17 = 97$$

$$\frac{x}{2} + 17 - 17 = 97 - 17$$ $$\frac{x}{2} = 80$$

$$\frac{x}{2} \times 2 = 80 \times 2$$ $$x = 160$$

The value of $x$ is $\boxed{\text{160}}$.

"Double a number, $y$, plus 4 is equal to -4" translates to:

$$2y + 4 = -4$$

$$2y + 4 - 4 = -4 - 4$$ $$2y = -8$$

$$\frac{2y}{2} = \frac{-8}{2}$$ $$y = -4$$

The value of $y$ is $\boxed{\text{-4}}$.

"Four times a number, $z$, is one more than 59" translates to:

$$4z = 59 + 1$$ $$4z = 60$$

$$\frac{4z}{4} = \frac{60}{4}$$ $$z = 15$$

The value of $z$ is $\boxed{\text{15}}$.

"The square of an integer, $p$, is 21 less than 121" translates to:

$$p^2 = 121 - 21$$ $$p^2 = 100$$

Remember that an integer can be positive or negative.

$$\sqrt{p^2} = \sqrt{100}$$ $$p = \pm 10$$

The value of $p$ is $\boxed{\text{10 or -10}}$.