If the system of linear equations [ aligned & 7 x+11 y+ z=13 & 5 x+4 y+7 z= & 175 x+194 y+…

Question

If the system of linear equations

$$
\begin{aligned}
& 7 x+11 y+\alpha z=13 \\
& 5 x+4 y+7 z=\beta \\
& 175 x+194 y+57 z=361
\end{aligned}
$$

has infinitely many solutions, then $$\alpha+\beta+2$$ is equal to :

Accepted Answer

Given,

$$
\begin{aligned}
& 7 x+11 y+\alpha z=13 \\
& 5 x+4 y+7 z=\beta \\
& 175 x+194 y+57 z=361
\end{aligned}
$$

$$
\text { For infinite solution, }\left|\[\begin{array}{ccc} 7 & 11 & \alpha \ 5 & 4 & 7 \ 175 & 194 & 57 \end{array}$$\right|=0
\]

$$
\Rightarrow\left|\[\begin{array}{ccc} 7 & 11 & \alpha \ 5 & 4 & 7 \ 0 & -81 & 57-25 \alpha \end{array}$$\right|=0
\]

$$
\begin{aligned}
& \Rightarrow 81(49-5 \alpha)+(57-25 \alpha)(-27)=0 \\
& \Rightarrow 270 \alpha=-2430 \Rightarrow \alpha=-9
\end{aligned}
$$

And $\Delta_1=0$

$$
\begin{aligned}
& \left|\[\begin{array}{ccc} 13 & 11 & -9 \ \beta & 4 & 7 \ 361 & 194 & 57 \end{array}\end{aligned}$$\right|=0 \\
\& \Rightarrow \beta=11

\]

$$
\therefore \alpha+\beta+2=4
$$

Solutions · 1