This assignment assesses your skills on solving the systems of linear and non-linear equations by various methods.

Linear and non-linear equations are helpful in solving optimization techniques and cost functions more specifically.

Understanding these types of equations allows us to describe and predict phenomena, optimize processes, design systems, and make informed decisions. Whether it’s calculating the trajectory of a projectile, modeling the growth of a population, or analyzing the behavior of financial markets, the ability to work with both linear and nonlinear equations is a powerful tool for solving real-world problems and advancing our understanding of the world around us.

You are required to complete all 3 tasks in this assignment. You can use scientific calculators or online scientific calculators for this assignment. When you are instructed to make a graph in this assignment, please use

GeoGebra graphing tool

. Show all steps wherever necessary.

**Task 1:**

Daniel takes pleasure in experimenting with various kinds of toys. He bought three distinct types of toys for $25 in total. Furthermore, he possesses a toy that costs $20 more than three times the combined price of the other two toys. It’s revealed that the total cost of the three toys multiplied by five, amounts to $125.

Using the above scenario, answer the following questions by using the necessary steps:

(i) Represent this situation in system of linear equations.

(ii) Identify the nature of the system (consistent, independent and number of solutions).

(iii) Find how much is the cost of each toy. Explain the answer clearly.

Task 2:

Find the solution of the following system of equations using Gaussian elimination method and show the steps.

x-y+z = 4 5x-y+z = 6 3x-y+5z= 5

Task 3:

Solve the following system of equations. Verify your answers algebraically and graphically.

Draw the graph and locate the points of intersections.