Determine the values of $a$ and $b$ for which the system $\left\{\begin{array}{l}2 x+y+a z=-1 \\ 3 x-2 y+z=b \\ 5 x-8 y+9 z=3\end{array}\right.$

Full solution

Q. Determine the values of

a

and

b

for which the system

\left\{\begin{array}{l}2 x+y+a z=-1 \\ 3 x-2 y+z=b \\ 5 x-8 y+9 z=3\end{array}\right.

Write Equations: Step $1$ : Write down the system of equations. \{[\(2x + y + az = $-1$ ], [ $3$ x - $2$ y + z = b], [ $5$ x - $8$ y + $9$ z = $3$ ]\}
Eliminate Z: Step $2$ : Use elimination to simplify the system. Start by eliminating $z$ from the first two equations. $\newline$ Multiply the first equation by $1$ and the second by $a$ to align coefficients of $z$ : $\newline$ $[2x + y + az = -1]$ $\newline$ $[3ax - 2ay + az = ab]$ $\newline$ Subtract the first from the second: $\newline$ $(3a - 2)x - (2a - 1)y = ab + 1$
Solve Simplified System: Step $3$ : Now, eliminate $z$ from the second and third equations. $\newline$ Multiply the second equation by $9$ and the third by $1$ : $\newline$ $[27x - 18y + 9z = 9b]$ $\newline$ $[5x - 8y + 9z = 3]$ $\newline$ Subtract the third from the first: $\newline$ $(27 - 5)x - (18 - 8)y = 9b - 3$ $\newline$ $22x - 10y = 9b - 3$
Solve Simplified System: Step $3$ : Now, eliminate $z$ from the second and third equations. $\newline$ Multiply the second equation by $9$ and the third by $1$ : $\newline$ $[27x - 18y + 9z = 9b]$ $\newline$ $[5x - 8y + 9z = 3]$ $\newline$ Subtract the third from the first: $\newline$ $(27 - 5)x - (18 - 8)y = 9b - 3$ $\newline$ $22x - 10y = 9b - 3$ Step $4$ : Solve the simplified system of equations from Steps $2$ and $3$ . $\newline$ From Step $2$ : $(3a - 2)x - (2a - 1)y = ab + 1$ $\newline$ From Step $3$ : $22x - 10y = 9b - 3$ $\newline$ Use substitution or elimination to find $a$ and $b$ .