Solving a System of Equations

System of Three Equations

• Given equations:
  • 2x + y + z = 7
  • 2x - y + 2z = 6
  • x - 2y + z = 0

Steps to Solve:

1. Choose Two Equations and Eliminate a Variable

   • Select the first two equations to eliminate y:
     • Add (2x + y + z) + (2x - y + 2z) = 7 + 6
     • Result: 4x + 3z = 13
   • Save this equation for later.

2. Combine Another Two Equations

   • Use the first and third equations:
   • Multiply the first equation by 2:
     • Resulting equation: 4x + 2y + 2z = 14
   • Add the modified first equation and the third equation:
     • Result: 5x + 3z = 14

3. Solve the Two Simplified Equations

   • Equations: 4x + 3z = 13 and 5x + 3z = 14
   • Subtract to eliminate z:
     • Multiply the first by -1 and add:
     • Result: x = 1
   • Substitute x = 1 into one of the equations:
     • 5(1) + 3z = 14 -> z = 3

4. Find y Using Original Equation

   • Substitute x = 1 and z = 3 into 2x + y + z = 7:
     • y + 5 = 7 -> y = 2

5. Result

   • Solution: x = 1, y = 2, z = 3

Solving a Word Problem Involving Investments

Problem Statement

• Two investments totaling $13,000 at 15% and 14% interest rates.
• Total interest received: $1,900.

Steps to Solve:

1. Define Variables

   • Let x = amount invested at 15%
   • Let y = amount invested at 14%
   • Equations:
     • x + y = 13,000
     • 0.15x + 0.14y = 1,900

2. Eliminate Decimals

   • Multiply the interest equation by 100:
     • 15x + 14y = 190,000

3. Solve Using Elimination

   • Multiply the first equation by -14:
   • Add to eliminate y:
     • x = 8,000 (amount invested at 15%)
   • y = 5,000 (remaining amount at 14%)

4. Verification

   • Calculate interests:
     • 15% of 8,000 = $1,200
     • 14% of 5,000 = $700
   • Total interest matches: $1,900

Solving a Problem Involving Cost of Fruits

Problem Statement

• Cost of apples and bananas:
  • 5 apples + 8 bananas = $6.55
  • 9 apples + 7 bananas = $9.20
• Find cost of 7 apples and 10 bananas.

Steps to Solve:

1. Define Variables

   • a = cost of one apple
   • b = cost of one banana

2. Write Equations

   • 5a + 8b = 6.55
   • 9a + 7b = 9.20

3. Eliminate a Variable

   • Multiply first equation by -7, second by 8:
   • Add equations to eliminate b:
   • Solve for a:
     • a = $0.75 (cost per apple)

4. Solve for b

   • Substitute a = 0.75 back into the first equation:
   • Solve for b:
     • b = $0.35 (cost per banana)

5. Calculate Cost of 7 Apples and 10 Bananas

   • 7 × $0.75 + 10 × $0.35 = $8.75

These steps solve the systems of equations and word problems presented, ensuring the verification of results through calculations.