The system of equations
\(2x + 3y = 13\)
\(4x - y = 5\)
is solved by Cramer’s rule. Then the determinant \(\Delta_y\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(-42\)
In Cramer’s rule, \(\Delta_y\) is the determinant of the coefficients with the y-column replaced by the constants 13 and 5.
\( \displaystyle \Delta_y = \begin{vmatrix} 2 & 13 \\ 4 & 5 \end{vmatrix} \)
\(\Delta_y = 2 \times 5 - 13 \times 4\)
\(\Delta_y = 10 - 52 = -42\)
Why the other options are incorrect
Option (1) \(42\) — Incorrect. This comes from \(52 - 10\), with the two diagonal products subtracted the wrong way round.
Option (2) \(-28\) — Incorrect. This is \(\Delta_x = 13 \times (-1) - 3 \times 5\), where the x-column is replaced.
Option (4) \(-14\) — Incorrect. This is \(\Delta = 2 \times (-1) - 3 \times 4\), the determinant of the coefficients alone.
With these values, \(x = (-28) \div (-14) = 2\) and \(y = (-42) \div (-14) = 3\).