Question

If the system of equations
\[ x - 2y + 3z = 9 \]
 \[2x + y + z = b \]
  \[x - 7y + az = 24 \] , has infinitely many solutions, then \[a - b\] is equal to:

Answer 1

Hint: In this question, we are asked to find the value of \[a-b\]. For that, we will write the system of linear equations in the determinant form, then find the value of \[D, D_{1}\] and substitute the values of a and b to find the required result.

Formula used:
We have been using the following formulas to find the determinant:
1.\[\begin{vmatrix}
 a_{11}&a_{12} &a_{13} \\
 a_{21}& a_{22} & a_{23} \\
a_{31} &a_{32} & a_{33} \\
\end{vmatrix}=a_{11}\left ( a_{22}a_{33}-a_{32}a_{23} \right )-a_{12}\left ( a_{21}a_{33}-a_{31}a_{23}\right )+a_{13}\left ( a_{21}a_{32}-a_{31}a_{22} \right )\]

Complete step-by-step solution:
We are given a system of linear equations is
\[x - 2y + 3z = 9 \]
\[2x + y + z = b \]
  \[x - 7y + az = 24\] , has infinitely many solutions
Now we will write the system of linear equations in determinant form, we get
\[D = \,\,\left| {\begin{array}{*{20}{c}}
  1&{ - 2}&3 \\
  2&1&1 \\
  1&{ - 7}&a
\end{array}} \right|\],
And we will replace first column with the terms of RHS of equations to get $D_1$.
\[{D_1} = \,\,\left| {\begin{array}{*{20}{c}}
  9&{ - 2}&3 \\
  b&1&1 \\
  {24}&{ - 7}&8
\end{array}} \right|\]
Now we know that the given system has infinitely many solutions so by the Cramer rule both determinants are equal to zero.
Now we find the determinant of \[D\], we have
\[
  \left| {\begin{array}{*{20}{c}}
  1&{ - 2}&3 \\
  2&1&1 \\
  1&{ - 7}&a
\end{array}} \right| = 0 \\
  1\left( {a - \left( { - 7} \right)} \right) + 2\left( {2a - 1} \right) + 3\left( { - 14 - 1} \right) = 0 \\
  1\left( {a + 7} \right) + 2\left( {2a - 1} \right) + 3\left( { - 15} \right) = 0 \\
  a + 7 + 4a - 2 - 45 = 0
 \]
Further solving, we get
\[
  5a + 5 - 45 = 0 \\
  5a - 40 = 0 \\
  5a = 40 \\
  a = 8
 \]
Now we find the determinant of \[{D_1}\], we have
\[
  \left| {\begin{array}{*{20}{c}}
  9&{ - 2}&3 \\
  b&1&1 \\
  {24}&{ - 7}&8
\end{array}} \right| = 0 \\
  9\left( {8 - \left( { - 7} \right)} \right) + 2\left( {8b - 24} \right) + 3\left( { - 7b - 24} \right) = 0 \\
  9\left( {8 + 7} \right) + 2\left( {8b - 24} \right) + 3\left( { - 7b - 24} \right) = 0 \\
  9\left( {15} \right) + 16b - 48 - 21b - 72 = 0 \]
Further solving, we get
\[
  135 - 48 - 72 - 5b = 0 \\
  15 - 5b = 0 \\
  15 = 5b \\
  b = 3
 \]
Therefore, the value of \[a - b\] is
\[
  a - b = 8 - 5 \\
   = 3
 \]
Hence, the value of \[a - b\] is \[3\]

Additional information: A system of linear equations is made up of two or more linear equations with two or more variables, so that all equations in the system are considered at the same time.

Note: One possibility for making a mistake in this type of problem is incorrectly converting the column or row while determining the determinant. It is necessary to understand which column or row should be converted by which process, as well as how to expand the determinant through any row or column for easy calculation.