Solve the system of equations: 7x + 6y = 10 3x + 5y = -3

System of Equations:

• Equation 1: \( 7x + 6y = 10 \)
• Equation 2: \( 3x + 5y = -3 \)

Step-by-step solution:

1. Step 1: Multiply Equation 1 by 3 and Equation 2 by 7
   Multiply Equation 1 by 3: \( 3(7x + 6y) = 3(10) \) which gives \( 21x + 18y = 30 \).
   Multiply Equation 2 by 7: \( 7(3x + 5y) = 7(-3) \) which gives \( 21x + 35y = -21 \).
2. Step 2: Subtract the modified Equation 2 from the modified Equation 1
   Subtract the second modified equation from the first modified equation:
   \( (21x + 18y) - (21x + 35y) = 30 - (-21) \)
   \( 21x + 18y - 21x - 35y = 30 + 21 \)
   \( -17y = 51 \)
3. Step 3: Solve for y
   Divide by -17: \( y = 51 / -17 \)
   \( y = -3 \)
4. Step 4: Substitute y back into Equation 1 to find x
   Substitute \( y = -3 \) into the original Equation 1: \( 7x + 6y = 10 \)
   \( 7x + 6(-3) = 10 \)
   \( 7x - 18 = 10 \)
5. Step 5: Solve for x
   Add 18 to both sides: \( 7x = 10 + 18 \)
   \( 7x = 28 \)
   Divide by 7: \( x = 4 \)

Answer: x = 4, y = -3