Solve the followings 1 (a + b+ c) (a2+ b2 + c2 – ab – bc – ca) 2 (5x3 – 7x2y– xy2 + 3y3) (5x + 3y)

Let's solve the system of equations:2x+5y=1x−3y=4

We can solve the system of equations using either the inverse method or Cramer's method.

Inverse Method:

1. Write the augmented matrix:(2​5​1 1​−3​4​)

2. Find the inverse of the coefficient matrix:(2​5​1 1​−3​4​)−1=(111​​113​​−111​ 111​​−111​​111​​)

3. Multiply the inverse matrix by the constant matrix:(111​​113​​−111​ 111​​−111​​111​​)(1 4​)=(1123​ −117​​)

4. The solution is:x=1123​,y=−117​

Cramer's Method:

1. Write the system of equations in Cramer's form:{2x+5y​=1 x−3y​=4​

2. Calculate the determinant of the coefficient matrix:D=det(2​5 1​−3​)=−11

3. Calculate the determinants of the numerator matrices:Dx​=det(1​5 4​−3​)=46Dy​=det(2​1 4​−3​)=−7

4. The solution is:x=DDx​​=−1146​=−1123​y=DDy​​=−11−7​=117​

Answer:

x=1123​,y=−117​

Additional Notes:

• The inverse method is generally more efficient for smaller systems of equations.
• Cramer's method is generally more efficient for larger systems of equations.
• Both methods can be used to solve systems of equations with non-zero determinants.
• If the determinant of the coefficient matrix is zero, then the system of equations is either inconsistent (no solution) or has infinitely many solutions.

I hope this helps! Let me know if you have any other questions.