Solve equations using both inverse and Cramer Methods. 2x + 5y = 1 x – 3y = 4

. 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​