Pengertian Sistem Persamaan Linear Tiga Variabel

Summary

This video provides a detailed explanation of the concept of a system of linear equations with three variables, a topic from mathematics. The instructor introduces the subject by explaining the basic definition of linear equations with two variables, before progressing to the more complex system involving three variables. Examples are used to demonstrate how to form mathematical models based on real-life scenarios, such as purchases of school supplies and food items. The video also covers solving these equations and recognizing systems with three variables, with practical problem-solving exercises to help viewers understand the concept more clearly.

Takeaways

• 😀 Introduction to the topic: The lesson covers systems of linear equations with three variables and inequalities with two variables.
• 😀 System of linear equations with three variables (SPLTV) involves three equations and three variables, typically represented as x, y, and z.
• 😀 Example scenario: Parit, Eri, and Dika buy school supplies, leading to a system of equations that is modeled with variables for the price of pencils, pens, and rulers.
• 😀 The basic form of a system of three linear equations involves equations like: a1x + b1y + c1z = d1.
• 😀 Understanding that the exponents of variables in a system of linear equations are always 1 (degree 1).
• 😀 For a system to be classified as linear in three variables, it must have three variables and all variables must have a degree of 1.
• 😀 Mathematical modeling for the first example involves representing purchases of pencils, pens, and rulers in terms of x, y, and z, resulting in the equations 2x + 3y + 2z = 19,500, x + 2y + z = 11,500, and x + 2y + 2z = 14,000.
• 😀 The second example involves Lina, Mel, and Nani buying baked goods, and it leads to a system of equations to model the prices of cakes, bread, and chocolates.
• 😀 To create the mathematical model, assign variables to the unknowns: 'a' for cake price, 'b' for bread price, and 'c' for chocolate price.
• 😀 To identify a valid system of linear equations, it’s important that there are three equations and the variables must be linear, i.e., raised only to the first power.

Q & A

What is a system of linear equations with three variables?

A system of linear equations with three variables is a set of three equations that each involve three variables, where each variable has a degree of 1. These systems can be represented in the form of equations like ax + by + cz = d.

How is a system of linear equations with three variables represented?

It is generally represented as: a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3, where x, y, and z are the variables, and the coefficients a1, b1, c1, d1, etc. are constants.

Can you explain the significance of the example given in the script with purchases of stationery items?

In the example, the problem presents purchases made by different individuals (Parit, Eri, and Dika) involving pencils, pens, and rulers. These are used to create a system of linear equations with three variables (x for pencil price, y for pen price, and z for ruler price), demonstrating how the system works in real-life scenarios.

What is the first equation derived from Parit's purchase in the example?

The first equation is 2x + 3y + 2z = 19,500, where x represents the price of a pencil, y the price of a pen, and z the price of a ruler.

What is the relationship between the variables in the system of equations presented?

Each variable (x, y, z) represents the price of an item (pencil, pen, or ruler), and the equations are formed based on the total cost of these items purchased in various combinations by different individuals.

How do we form a system of linear equations with three variables from a word problem?

We translate the word problem into mathematical equations by identifying what each variable represents (e.g., prices of items) and then expressing the total cost of the items in terms of those variables.

What is the importance of understanding the degree of variables in linear equations?

In linear equations, the degree of each variable must be 1. This ensures that the relationship between variables is linear. If any variable has a degree higher than 1, the equation no longer remains linear.

What type of system of equations is being discussed in the example from the script?

The script discusses a system of linear equations with three variables. The equations are linear because each variable appears to the first power (degree 1).

Why is the equation x + 2y + 2z = 14,000 important in the example?

This equation represents the total cost of the items purchased by Dika, with the prices of the items (pencil, pen, ruler) denoted by x, y, and z. It is one of the key equations in the system used to solve for the prices.

How do you determine if a system of equations involves three variables?

A system involves three variables if each equation includes exactly three different variables. For example, equations like 2x + 3y + 2z = 19,500 have three variables (x, y, z), which is a defining characteristic of a system of three-variable linear equations.