Sistem Persamaan Linear Tiga Variabel (SPLTV) | Matematika Wajib Kelas X

Summary

In this video, the topic of solving a system of three-variable linear equations is explored using the elimination and substitution methods. The presenter introduces the general form of a three-variable linear system and guides viewers through the process of eliminating variables, starting with equations 1 and 2, followed by equations 1 and 3. The procedure includes finding the least common multiple (LCM) to align coefficients, performing substitutions, and ultimately solving for each variable. The result is presented as a solution set of x = 2, y = 3, and z = 4, demonstrating the steps clearly and effectively for students to follow.

Takeaways

• 😀 Introduction to the topic: The script discusses solving a system of linear equations with three variables.
• 😀 The general form of a system of three linear equations is presented as: A1x + B1y + C1z = D1, A2x + B2y + C2z = D2, A3x + B3y + C3z = D3.
• 😀 The objective is to find the values of three variables: x, y, and z in the system of equations.
• 😀 Three main methods for solving the system of linear equations are introduced: elimination, substitution, and a combination of both.
• 😀 The script explains the process of eliminating one variable by matching coefficients of variables.
• 😀 The Least Common Multiple (LCM) of 2 and 3 is used to eliminate the variable 'x' from the first and second equations.
• 😀 After elimination, a new equation is formed, which simplifies the system into two variables: y and z.
• 😀 The process of eliminating the variables continues with another set of equations (first and third equations), again using LCM.
• 😀 After obtaining two equations with two variables (y and z), elimination is performed once more to solve for z.
• 😀 After finding the value of z (z = 4), substitution is used to find the values of y and x by plugging z into the previous equations.
• 😀 The final solution is obtained as: x = 2, y = 3, z = 4, and this is written as the solution set { (x, y, z) = (2, 3, 4) }.

Q & A

What is a system of linear equations with three variables?

A system of linear equations with three variables consists of three equations, each involving three variables (x, y, and z). These equations are typically expressed in the form of Ax + By + Cz = D, where A, B, and C are coefficients, and D is a constant.

What methods can be used to solve a system of three-variable linear equations?

The methods that can be used to solve a system of three-variable linear equations are elimination, substitution, and a combination of both elimination and substitution.

What is the process of elimination in solving a system of equations?

Elimination involves removing one of the variables by making the coefficients of that variable in two equations equal. This is achieved by multiplying the equations by appropriate factors to make the coefficients match, then adding or subtracting the equations to eliminate the chosen variable.

How do you eliminate variables in a system of equations?

To eliminate a variable, you first identify the least common multiple (LCM) of the coefficients of the variable you want to eliminate. Then, you multiply the equations to make the coefficients match and add or subtract the equations to cancel out the variable.

What is the significance of the least common multiple (LCM) in the elimination process?

The least common multiple (LCM) of the coefficients of a variable in two equations is used to make the coefficients equal so that the variable can be eliminated by addition or subtraction. This simplifies the system and reduces the number of variables.

What is the next step after eliminating one variable from the system?

After eliminating one variable, you are left with a system of two equations in two variables. You can then proceed by using either substitution or elimination again to solve for the remaining variables.

What does the term 'himpunan penyelesaian' mean in the context of solving linear equations?

'Himpunan penyelesaian' refers to the solution set of the system of equations, which is the set of values for the variables (x, y, z) that satisfy all the equations in the system.

What is the final step in solving a system of linear equations using elimination and substitution?

The final step is to substitute the values obtained for two variables into one of the original equations to find the value of the remaining variable. Once all variables are solved, the solution set is written as a tuple (x, y, z).

How do you perform substitution in solving a system of equations?

Substitution involves replacing one of the variables in an equation with an expression derived from another equation. This allows you to solve for one variable at a time, progressively reducing the system until all variables are determined.

Why is it important to express the solution set in a specific format, such as (x, y, z)?

Expressing the solution set as (x, y, z) is important because it clearly indicates the values of each variable that satisfy the system of equations, providing a structured and standard way to represent the solution.