Solve the system x1 - 3x3 = 8, 2x1 + 2x2 + 9x3 = 7, and x2 + 5x3 = -2.

Step-by-step solution

1. 1

   Exploit the missing variables instead of grinding through elimination. Equation 1 contains no $x_2$ and equation 3 contains no $x_1$, so each one already expresses a variable in terms of $x_3$ alone. That makes substitution far cheaper than row reduction here.

2. 2

   Solve the two sparse equations for x1 and x2.

   $x_1=8+3x_3,\qquad x_2=-2-5x_3.$

3. 3

   Substitute both into the remaining equation.

   $2(8+3x_3)+2(-2-5x_3)+9x_3=7.$

4. 4

   Expand and collect.

   $16+6x_3-4-10x_3+9x_3=7\ \Longrightarrow\ 12+5x_3=7\ \Longrightarrow\ x_3=-1.$

5. 5

   Back-substitute for the other two unknowns.

   $x_1=8+3(-1)=5,\qquad x_2=-2-5(-1)=3.$

6. 6

   Verify all three equations. $5-3(-1)=5+3=8$ ✓; $2(5)+2(3)+9(-1)=10+6-9=7$ ✓; $3+5(-1)=-2$ ✓. The coefficient determinant is $-5\neq 0$, so $(5,3,-1)$ is the unique solution.