Sistem Persamaan Linier Tiga Variabel (SPLTV), Soal Cerita

Introduction

This tutorial provides a step-by-step guide on solving a system of linear equations with three variables (SPLTV) using a story problem. The example involves finding the price of stationery items based on given equations. Understanding this method will enhance your problem-solving skills in algebra and improve your ability to tackle similar word problems.

Step 1: Formulate the Equations

Begin by translating the problem statement into mathematical equations.

1. Identify the variables:

   • Let x be the price of one notebook.
   • Let y be the price of one pencil.
   • Let z be the price of one ruler.

2. Write down the equations based on the information provided:

   • Equation 1: ( 5x + 3y + 2z = 36,500 )
   • Equation 2: ( 2x + 4y + 1z = 25,000 )
   • Equation 3: ( 1x + 1y + 3z = 16,500 )

Step 2: Use the Elimination Method

To solve the equations, we will use the elimination method to eliminate one variable at a time.

1. Start with Equations 1 and 2. Multiply Equation 2 by a factor that allows the elimination of one variable when combined with Equation 1:

   • Multiply Equation 2 by 2:
     • ( 4x + 8y + 2z = 50,000 )

2. Subtract the adjusted Equation 2 from Equation 1:

   • ( (5x + 3y + 2z) - (4x + 8y + 2z) = 36,500 - 50,000 )
   • This simplifies to:
     • ( x - 5y = -13,500 )
   • We can label this as Equation 4.

Step 3: Solve for Variables

Next, use Equation 4 along with one of the original equations to find the values of the variables.

1. Use Equation 4 and Equation 3:

   • From Equation 3, replace ( x ) from Equation 4:
     • Substitute ( x = 5y - 13,500 ) into Equation 3:
     • ( (5y - 13,500) + y + 3z = 16,500 )

2. Combine like terms:

   • ( 6y + 3z - 13,500 = 16,500 )
   • Simplifying gives:
     • ( 6y + 3z = 30,000 )
   • Divide by 3:
     • ( 2y + z = 10,000 )
   • Label this as Equation 5.

Step 4: Substitute and Solve

1. Now use Equation 5 to express ( z ) in terms of ( y ):

   • ( z = 10,000 - 2y )

2. Substitute ( z ) back into any of the previous equations (e.g., Equation 2):

   • Substitute into ( 2x + 4y + z = 25,000 ):
   • ( 2x + 4y + (10,000 - 2y) = 25,000 )
   • This simplifies to:
     • ( 2x + 2y = 15,000 )
   • Divide everything by 2:
     • ( x + y = 7,500 )
   • Label this as Equation 6.

Step 5: Solve for Each Variable

1. Now you have:

   • From Equation 4: ( x - 5y = -13,500 )
   • From Equation 6: ( x + y = 7,500 )

2. Use substitution or elimination to solve for ( y ) and then ( x ):

   • Substitute ( x = 7,500 - y ) into Equation 4:
   • ( (7,500 - y) - 5y = -13,500 )
   • Simplifying gives:
     • ( 7,500 - 6y = -13,500 )
     • ( 6y = 21,000 )
     • ( y = 3,500 )

3. Substitute ( y ) back to find ( x ) and ( z ):

   • From Equation 6: ( x + 3,500 = 7,500 ) gives ( x = 4,000 ).
   • Substitute ( y ) into Equation 5:
   • ( z = 10,000 - 2(3,500) = 3,000 ).

Conclusion

You have successfully solved for the prices of the items:

• A notebook costs Rp4,000.
• A pencil costs Rp3,500.
• A ruler costs Rp3,000.

This process illustrates how to approach a system of linear equations through formulation, elimination, and substitution. You can apply these steps to other similar word problems for effective solutions.