Sistem Persamaan Linear Tiga Variabel Matematika Wajib Kelas 10 Bagian 1

Introduction

In this tutorial, we will explore how to solve systems of linear equations with three variables, a key topic in mathematics for 10th-grade students. We will utilize two primary methods: elimination and substitution. Understanding these methods will enhance your problem-solving skills and prepare you for more advanced mathematical concepts.

Step 1: Understanding the System of Equations

Before diving into solving the equations, it's essential to grasp what a system of linear equations is.

• A system of linear equations consists of multiple linear equations that share the same variables.
• In this case, we will focus on three equations with three variables (x, y, z).

Example Equations:

1. 2x + 3y + z = 1
2. 4x + y - z = 2
3. -x + 2y + 5z = 3

Step 2: Using the Elimination Method

The elimination method involves eliminating one variable at a time to simplify the system into a solvable form.

1. Choose a variable to eliminate. For instance, let's eliminate z.

2. Align the equations for easy comparison.

3. Manipulate the equations to eliminate z:

   • From equations 1 and 2, manipulate them to express one equation without z.
   • For example:
     • Add equations 1 and 2 to eliminate z:

       (2x + 3y + z) + (4x + y - z) = 1 + 2
       

       Resulting in:

       6x + 4y = 3  [Equation 4]
       

4. Repeat the process with the remaining equations to form a new system with two variables.

Step 3: Solving the Reduced System

Once you have a system with two variables, solve it using the same elimination method or substitution.

1. Choose one of the new equations formed in Step 2.

2. Isolate one variable:

   • For example, from Equation 4:

     6x + 4y = 3
     

     Isolate y:

     4y = 3 - 6x
     y = (3 - 6x) / 4
     

3. Substitute into another equation from the reduced system to find the value of x.

Step 4: Finding the Third Variable

After solving for two variables (x and y), substitute back into one of the original equations to find the value of the third variable, z.

1. Use one of the original equations:

   2x + 3y + z = 1
   

2. Substitute the values of x and y obtained from the previous steps to solve for z.

Conclusion

In this tutorial, we covered the steps to solve a system of linear equations with three variables using both elimination and substitution methods. Practice these steps with various sets of equations to strengthen your understanding.

To continue your learning, consider watching the next video in the series for more complex examples and applications. Happy solving!