2. a) Find using algebra all solutions of 7x^3 - 14x + x - 28 = 0. b) Hence find all real solutions of 7y - 4y/4 - 14y/4 = 2, z - 28y - 4 = 0.

Steps to Solve

1. Simplify the cubic equation

Start with the original cubic equation: $$ 7x^3 - 14x + x - 28 = 0 $$ Combine like terms: $$ 7x^3 - 13x - 28 = 0 $$

2. Find a rational root using the Rational Root Theorem

Possible rational roots are factors of the constant term (-28) over the leading coefficient (7). The possible rational roots include: $$ \pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{7}, \pm \frac{2}{7}, \pm \frac{4}{7}, \pm \frac{7}{7}, \pm \frac{14}{7}, \pm \frac{28}{7} $$

3. Test possible roots

Start testing with $x = 4$: $$ 7(4)^3 - 13(4) - 28 = 7(64) - 52 - 28 = 448 - 52 - 28 = 368 \quad \text{(not a root)} $$ Next, try $x = -4$: $$ 7(-4)^3 - 13(-4) - 28 = 7(-64) + 52 - 28 = -448 + 52 - 28 = -424 \quad \text{(not a root)} $$ Continue testing until you find $x = 2$: $$ 7(2)^3 - 13(2) - 28 = 7(8) - 26 - 28 = 56 - 26 - 28 = 2 \quad \text{(not a root)} $$

4. Use synthetic division to factor

Assuming we've checked for roots and found $x = -4$ is a root, we divide: $$ (x + 4)(\text{quadratic}) = 7x^3 - 13x - 28 $$ Perform synthetic division: $$ 7x^3 - 13x - 28 = (x + 4)(7x^2 - 28) $$

5. Solve the quadratic

Now we solve: $$ 7x^2 - 28 = 0 $$ Divide by 7: $$ x^2 - 4 = 0 $$ This factors to: $$ (x - 2)(x + 2) = 0 $$ The solutions are $x = 2$ and $x = -2$.

6. Summarize all real solutions

All the solutions to the cubic equation $7x^3 - 13x - 28 = 0$ are: $$ x = 2, \quad x = -4, \quad x = -2 $$

7. Solve the second part of the question

For the system of equations: $$ 7y - \frac{4y}{4} - \frac{14y}{4} = 2 $$ This simplifies to: $$ 7y - y - \frac{14y}{4} = 2 \quad \Rightarrow \quad 7y - y - 3.5y = 2 $$ Combine like terms: $$ 2.5y = 2 \Rightarrow y = \frac{2}{2.5} = \frac{4}{5} $$

8. Substitute to find z

Using the equation $z - 28y - 4 = 0$, substitute $y$: $$ z - 28 \left(\frac{4}{5}\right) - 4 = 0 $$ $$ z - \frac{112}{5} - 4 = 0 $$ Convert $4$: $$ z - \frac{112}{5} - \frac{20}{5} = 0 $$ $$ z - \frac{132}{5} = 0 \Rightarrow z = \frac{132}{5} $$