SISTEM PERSAMAAN LINIER TIGA VARIABEL

Summary

In this lesson, we explore solving systems of linear equations with three variables, or SPLTV. The script breaks down essential concepts such as variables, coefficients, and constants in linear equations. Using examples, the teacher demonstrates methods like elimination and substitution to find values for variables (x, y, z) in different systems of equations. Through step-by-step calculations, learners are guided on how to manipulate equations and solve for unknowns, ensuring a thorough understanding of these critical algebraic techniques. The lesson concludes with practical exercises and encourages students to apply these methods for problem-solving.

Takeaways

• 😀 Linear equations involve variables, coefficients, and constants. Variables are usually represented by letters from a to z, coefficients are the numbers attached to variables, and constants are numbers without variables.
• 😀 The general form of a linear equation is AX + BY + CZ = D, where A, B, and C are coefficients and X, Y, Z are variables.
• 😀 A linear equation must have a degree of one, meaning it doesn't have exponents on the variables (no squared or cubed terms).
• 😀 A system of linear equations with three variables involves solving multiple equations to find the values of X, Y, and Z.
• 😀 There are three main methods to solve systems of linear equations: elimination, substitution, and a combination of both.
• 😀 In the elimination method, you eliminate one variable by combining two equations in such a way that one of the variables cancels out.
• 😀 The substitution method involves solving one of the equations for one variable and then substituting that value into the other equations.
• 😀 In the example, the elimination method is applied between two equations to eliminate the Y variable and solve for X and Z.
• 😀 After eliminating variables, substitution is used to find the specific values of the variables in the system (X, Y, Z).
• 😀 The final solution of the system of equations is the set of values for X, Y, and Z that satisfy all the equations in the system.
• 😀 The process of solving a system of linear equations can involve several steps: eliminating variables, substituting values, and simplifying to find the solution.

Q & A

What is a linear equation with three variables?

A linear equation with three variables is an equation where the highest power of the variables is 1. It involves three variables (typically x, y, and z), and the equation is in the form of AX + BY + CZ = D, where A, B, C, and D are constants.

What are the components of a linear equation?

A linear equation consists of three main components: variables (usually denoted as x, y, z, etc.), coefficients (the numerical multipliers of the variables), and constants (numbers that do not have variables associated with them).

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

'Linear' refers to the fact that the variables in the equation are to the first power only, meaning there are no exponents or powers greater than 1 for the variables.

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

A system of linear equations with three variables is formed when multiple linear equations with the same three variables are combined together. For example, a system could have equations like x + y + z = -4, 2x + 5y + z = 6, and 9x + 8y + 5z = 7.

What are the methods to solve a system of linear equations with three variables?

There are three common methods to solve such a system: elimination, substitution, and a combination of both methods (called the elimination-substitution method).

What is the elimination method in solving systems of linear equations?

The elimination method involves adding or subtracting equations in such a way that one of the variables is eliminated. This simplifies the system, allowing the remaining equations to be solved for the other variables.

How does the substitution method work for solving a system of linear equations?

The substitution method involves solving one of the equations for one variable and then substituting this expression into the other equations. This process reduces the system to simpler equations that are easier to solve.

Can you describe a step from the example using the elimination method?

In the example, the first step involves combining two equations in the system and eliminating the variable y by adding them. After simplifying, you are left with an equation involving only x and z, which can then be solved.

What does it mean to have a system that has a solution, and how is it represented?

A system has a solution when there is a unique set of values for the variables (x, y, z) that satisfies all equations in the system. The solution is represented as an ordered triplet, such as (x, y, z) = (-2, 1, -1).

How does the method of substitution help when solving for variables?

Substitution helps by isolating one variable in one of the equations and then replacing it in the other equations. This makes it easier to reduce the system to a simpler form and solve for the remaining variables.