Notes on Systems of Linear Equations in Two Variables

Objective

• Solve problems involving systems of linear equations in two variables.

Example 1: Sum and Difference of Two Numbers

• Problem Statement: The sum of two numbers is 10 and the difference is 46. Find the numbers.
  • Understand the problem:
    • Sum: 10
    • Difference: 46
• Variables:
  • Let ( x ) = larger number
  • Let ( y ) = smaller number

Step 1: Formulate Equations

• Equation 1: ( x + y = 10 )
• Equation 2: ( x - y = 46 )

Step 2: Choose Method

• Use the Elimination Method.

Step 3: Solve the System

• Add the two equations:
  ( (x + y) + (x - y) = 10 + 46 )
  • Result: ( 2x = 56 )
  • Solve for ( x ): ( x = 28 )

Step 4: Substitute to Find ( y )

• Substitute ( x ) in Equation 1:
  ( 28 + y = 10 )
  • Result: ( y = 10 - 28 )
  • So, ( y = -18 )

Step 5: Check the Solution

• Check in both equations:
  • For Equation 1: ( 28 + (-18) = 10 ) (True)
  • For Equation 2: ( 28 - (-18) = 46 ) (True)
• Final Solution: The two numbers are 28 and -18.

Example 2: Carpenter's Wood Problem

• Problem Statement: A carpenter wants to cut a 16-foot piece of wood into two pieces. The longer piece is to be one foot longer than twice the shorter piece. Find the lengths.

Step 1: Understand the Problem

• Total length: 16 feet
• Let ( x ) = length of shorter piece
• Let ( y ) = length of longer piece

Step 2: Formulate Equations

• Equation 1: ( x + y = 16 )
• Equation 2: ( y = 2x + 1 )

Step 3: Solve Using Substitution

• Substitute Equation 2 into Equation 1:
  ( x + (2x + 1) = 16 )
  • Combine like terms: ( 3x + 1 = 16 )
  • Solve for ( x ): ( 3x = 15 )
  • Result: ( x = 5 )

Step 4: Find ( y )

• Substitute ( x ) into Equation 2:
  ( y = 2(5) + 1 )
  • Result: ( y = 11 )

Step 5: Check the Solution

• Check in both equations:
  • For Equation 1: ( 5 + 11 = 16 ) (True)
  • For Equation 2: ( 11 = 2(5) + 1 ) (True)
• Final Solution: Shorter piece = 5 feet, Longer piece = 11 feet.

Example 3: Saving Coins Problem

• Problem Statement: Alex saved 20 coins consisting of 25-cent and 1-peso coins. The total amount is 10.25 pesos.

Step 1: Understand the Problem

• Let ( x ) = number of 25-cent coins
• Let ( y ) = number of 1-peso coins

Step 2: Formulate Equations

• Equation 1: ( x + y = 20 )
• Equation 2: ( 0.25x + 1y = 10.25 )

Step 3: Use Elimination Method

• Multiply Equation 1 by -1: ( -x - y = -20 )

Step 4: Solve the System

• Combine with Equation 2:
  • Result: ( -y + 0.25y = -20 + 10.25 )
  • Solve for ( x ): ( x = 13 )

Step 5: Find ( y )

• Substitute ( x ) into Equation 1:
  ( 13 + y = 20 )
  • Result: ( y = 7 )

Step 6: Check the Solution

• Check in both equations:
  • For Equation 1: ( 13 + 7 = 20 ) (True)
  • For Equation 2: ( 0.25(13) + 7 = 10.25 ) (True)
• Final Solution: Alex has 13 pieces of 25-cent coins and 7 pieces of 1-peso coins.

Conclusion

• Systems of linear equations can be solved using various methods such as elimination and substitution. Always check the solution against the original equations to ensure accuracy.