To solve the system of equations using substitution, we will proceed step by step.

Given System:

1. $x + y = 0$
2. $x^3 - 17x - y = 0$

Step 1: Express $y$ in terms of $x$ from the first equation.

From $x + y = 0$, we get: $y = -x$

Step 2: Substitute $y = -x$ into the second equation.

Substitute $y = -x$ into $x^3 - 17x - y = 0$: $x^3 - 17x - (-x) = 0$ $x^3 - 17x + x = 0$ $x^3 - 16x = 0$

Step 3: Factorize $x^3 - 16x = 0$.

$x(x^2 - 16) = 0$ $x(x - 4)(x + 4) = 0$

This gives the solutions: $x = 0, \, x = 4, \, x = -4$

Step 4: Find the corresponding $y$-values.

Using $y = -x$:

• For $x = 0$, $y = -0 = 0$. So, $(x, y) = (0, 0)$.
• For $x = 4$, $y = -4$. So, $(x, y) = (4, -4)$.
• For $x = -4$, $y = 4$. So, $(x, y) = (-4, 4)$.

Final Solutions:

1. Smallest $x$-value: $(-4, 4)$
2. Middle $x$-value: $(0, 0)$
3. Largest $x$-value: $(4, -4)$

Would you like me to graph these solutions or explain further?

Relative Questions:

1. How do we use substitution to simplify a system of equations?
2. Why is it important to check solutions graphically?
3. How do cubic equations determine the number of intersections in a system?
4. What is the significance of factoring when solving polynomial equations?
5. How do you interpret the relationship between $x$ and $y$ in this system?

Tip: Always verify your solutions graphically or by substituting back into the original equations.