Sistem Persamaan Linear Tiga Variabel Matematika Wajib Kelas 10 Bagian 2

Introduction

This tutorial aims to guide you through solving systems of linear equations with three variables, particularly focusing on equations that are in fractional form with variables as denominators. Additionally, we will explore how to tackle contextual problems or word problems involving these equations, as discussed in the video from m4th-lab.

Step 1: Understanding the Structure of Linear Equations

To solve linear equations with three variables, it's essential to understand their structure:

A typical linear equation in three variables (x, y, z) can be represented as

• Ax + By + Cz = D
• Here, A, B, C, and D are constants.

Practical Tips

• Ensure all equations are in standard form for easier manipulation.
• If variables are in the denominators, multiply through by the least common multiple (LCM) to eliminate fractions.

Step 2: Multiplying to Eliminate Fractions

When dealing with equations that have variables in the denominators:

1. Identify the denominators in each equation.
2. Calculate the LCM of these denominators.
3. Multiply each term in the equation by this LCM to eliminate fractions.

Example

If you have the equation:

• ( \frac{x}{2} + \frac{y}{3} = 5 )

Multiply through by 6 (the LCM of 2 and 3):

• ( 6 \cdot \frac{x}{2} + 6 \cdot \frac{y}{3} = 6 \cdot 5 )
• This simplifies to: ( 3x + 2y = 30 )

Step 3: Setting Up the System of Equations

Once you have eliminated fractions from all the equations:

1. Write down all equations in a clear format.
2. Ensure all equations are aligned to facilitate easy comparison and calculations.

Example

For three equations:

• ( 3x + 2y - z = 30 )
• ( 4x + 3y + 2z = 50 )
• ( x - y + z = 10 )

Step 4: Solving the System Using Substitution or Elimination

Choose either substitution or elimination to solve the system:

Substitution Method

1. Solve one of the equations for one variable.
2. Substitute this expression into the other equations.
3. Repeat until all variables are determined.

Elimination Method

1. Add or subtract equations to eliminate one variable.
2. Solve the resulting two-variable system.
3. Substitute back to find the third variable.

Step 5: Applying the Solution to Contextual Problems

When solving word problems:

1. Read the problem carefully to identify the variables.
2. Formulate equations based on the context provided.
3. Follow the steps outlined previously to solve for the variables.

Practical Tips

• Clearly define what each variable represents.
• Check your solution by substituting back into the original problem.

Conclusion

In this tutorial, we covered the process of solving systems of linear equations with three variables, including how to handle fractional forms and contextual problems. By following these structured steps, you can confidently approach similar mathematical problems. Practice with different equations to strengthen your skills, and don't hesitate to reach out for help if needed. Happy learning!