Sistem Persamaan Linear Tiga Variabel

Introduction

In this tutorial, we will learn how to solve a system of three linear equations with three variables using the elimination and substitution methods. This guide is designed to help students understand and apply these techniques not just in academic settings but also in real-life situations where such equations are relevant.

Step 1: Understand the System of Equations

Before solving, familiarize yourself with the format of a system of three linear equations, which typically looks like this:

1. a₁x + b₁y + c₁z = d₁
2. a₂x + b₂y + c₂z = d₂
3. a₃x + b₃y + c₃z = d₃

Where:

• x, y, and z are the variables.
• a₁, b₁, c₁, d₁, etc., are constants.

Practical Tips

• Ensure all equations are in standard form.
• Check for consistency and dependency among the equations.

Step 2: Use the Elimination Method

The elimination method involves eliminating one variable at a time to reduce the system to two equations with two variables.

1. Choose a variable to eliminate (for example, z).
2. Modify the equations to align coefficients:
   • Multiply equations as needed to make the coefficients of z equal.
3. Subtract or add equations:
   • Combine the modified equations to eliminate z, resulting in two new equations with x and y.
4. Repeat the elimination process for the new set of equations.

Common Pitfalls to Avoid

• Always double-check your arithmetic when modifying equations.
• Ensure you correctly apply the operations to both sides of the equations.

Step 3: Solve the Reduced System

Once you have two equations with two variables, solve them using either substitution or elimination.

1. Substitution Method:

   • Solve one of the equations for one variable (e.g., solve for y).
   • Substitute that expression into the other equation.
   • Solve for the remaining variable.

2. Elimination Method (optional):

   • Repeat the elimination process to solve for the two variables.

Example

If after eliminating z, you have:

1. 2x + 3y = 5
2. 4x - y = 11

You can solve for y in the first equation:

y = (5 - 2x) / 3

Then substitute y into the second equation.

Step 4: Back Substitute to Find Remaining Variables

Once you have the values for two variables, substitute them back into one of the original equations to find the third variable (z).

Example

Using the values found for x and y, substitute them back into:

a₁x + b₁y + c₁z = d₁

Solve for z.

Conclusion

In summary, solving a system of three linear equations involves understanding the equations' structure, using the elimination method to simplify to two variables, and then solving for all three variables. Practicing these techniques will enhance your problem-solving skills in mathematics and help you tackle real-world problems effectively. As a next step, consider practicing with different sets of equations to strengthen your understanding.