Animated Solution for Mathematics - Matrices and Determinants: If the system of linear equations $7x+11y+az=13$, $5x+4y+7z=\beta$, $175x+194y+57z=361$ has infinitely many solutions, then $a+\beta +2$ is equal to

When you see a system like $7x+11y+az=13$, $5x+4y+7z=\beta$, and $175x+194y+57z=361$, do not see just numbers. See three planes in three-dimensional space.

Usually, three planes intersect at a single point, giving us a unique solution. But here, we are told the system has infinitely many solutions. This means these planes are dancing in perfect harmony, intersecting along a single, shared line.

Cramer's Rule

To unlock this, we turn to the elegant machinery of Cramer's Rule. For a system to have infinitely many solutions, the main determinant of the coefficient matrix, $D$, must vanish.

But that is not enough! We also require the auxiliary determinants $D_1,D_2,D_3$ to be zero. Let us construct our main determinant:

Now, look at that third row: $175,194,57$. It looks intimidating, but look closer. $175$ is exactly $25\times 7$. This is a gift!

By applying the row operation $R_3\to R_3-25R_1$, we transform the determinant into something far more manageable:

Suddenly, the complexity melts away. Expanding along the third row, we get:

Dividing by $27$, we find $3(49-5a)-(57-25a)=0$, which simplifies to $147-15a-57+25a=0$. This leads us directly to $10a=-90$, or $a=-9$.

With $a=-9$ in hand, we turn to $D_1=0$ to find $\beta$. Replacing the first column with the constants $13,\beta ,361$, we get:

Expanding this, we find a common factor of $113$, which simplifies the equation to $13(-10)-21\beta +361=0$. This yields $231-21\beta =0$, so $\beta =11$.

Finally, the question asks for $a+\beta +2$. Substituting our values, we get $-9+11+2=4$.

You have navigated the geometry and the algebra with precision. The final result is $4$.