Answer to Question #194537 in Linear Algebra for Zeeshan

Question

Find the solution of the following system of equation by substitution method.

2x + 5y + 2z = -38

3x - 2y + 4z = 17

-6x + y - 7z = -12

Answer 1

Step 1: Solve for one variable (x) in equation 1

• 2x + 5y + 2z = -38
• 2x = -5y - 2z - 38
• x = -2.5y - z - 19

Step 2: Replace the value of x in the remaining 2 equations and solve by normal algebraic methods.

• 3(2.5y - z - 19) - 2y + 4z = 17
• -6(2.5y - z - 19) + y - 7z = -12
• Solving gives:
• 7.5y - 3z - 57 - 2y + 4z = 17
• -15y +6z + 114 + y - 7z = -12
• Collect like terms
• 5.5y + z = 74
• -14y - z = -126

Step 3: Solve for one variable (z) in one of the two equations in step 2 above

• z = 14y + 126

Step 4: Replace the value of z in the other equation in step 2 and solve by normal algebraic methods.

• 5.5y + 14y + 126 = 74
• Solving gives
• 19.5y = -52
• y = -8/3

Step 5: Replace the value of y in one of the equations in step 2 and find the value of z

• 5.5(-8/3) + z = 74
• z=266/3

Step 6: Knowing y and z, we can calculate the value of x in step 1

• x = -2.5y - z - 19
• x = -2.5(-8/3) - 266/3 - 19
• x = -101

Hence x = -101, y = -8/3 or (2²/₃), z = 266/3 or (88²/₃)

You can confirm if these are the correct answers by replacing them in the original equations and comparing the answers with the ones in the equation.

For instance, 2x + 5y + 2z = -38

2(-101) + 5(-8/3) + 2(266/3) will give -38.