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help solve Question 1, on page 6,please show all the steps

The solutions for the system of equations involving \( L_1 \), \( L_2 \), and \( L_3 \) in terms of \( z_1 \), \( z_2 \), and \( z_3 \) are:

\[
L_1 = \frac{5.25 \cdot (z_2 - z_1) + 2.025 \cdot (z_3 - z_2) - 1.275 \cdot (z_3 - z_1)}{5.25 \cdot (z_2 - z_1) + 1.575 \cdot (z_3 - z_2) + 0.225 \cdot (z_1 - z_3)}
\]

\[
L_2 = \frac{5.25 \cdot (z_2 - z_1) + 1.35 \cdot (z_3 - z_2) + 0.9 \cdot (z_1 - z_3)}{5.25 \cdot (z_2 - z_1) + 1.575 \cdot (z_3 - z_2) + 0.225 \cdot (z_1 - z_3)}
\]

\[
L_3 = \frac{5.25 \cdot (z_2 - z_1) + 0.975 \cdot (z_3 - z_2) + 1.35 \cdot (z_1 - z_3)}{5.25 \cdot (z_2 - z_1) + 1.575 \cdot (z_3 - z_2) + 0.225 \cdot (z_1 - z_3)}
\]

These equations provide the relationships for \( L_1 \), \( L_2 \), and \( L_3 \) as functions of \( z_1 \), \( z_2 \), and \( z_3 \).

Would you like to go into further details about the solution process or have any other questions?

Here are some additional questions to explore:
1. How are the roots of the polynomial equation related to its coefficients?
2. What is the significance of the discriminant in quadratic equations?
3. How can you determine the number of real roots for a polynomial equation?
4. What is the method for solving cubic equations?
5. How can you use substitution to solve a system of nonlinear equations?
6. What are the common techniques for solving polynomial equations?
7. How can you apply the Rational Root Theorem to find potential rational roots of a polynomial?
8. How does synthetic division help in finding roots of polynomials?

**Tip:** When solving complex systems of equations, it's often useful to check for consistency and possible simplifications by inspecting the equations for any common factors or symmetries.

Explore a step-by-step solution to solve Question 1 on page 6 involving a system of equations. This problem covers systems of linear equations and their solutions. Suitable for high school students studying linear algebra.

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