Sure, let's solve the given system of linear equations step by step and find [tex]\( x - y \)[/tex].
The system of equations is:
[tex]\[
\left\{\begin{array}{l}
2x + 5y = 31 \\
3x - 2y = -1
\end{array}\right.
\][/tex]
We can solve this system using the method of determinants (Cramer's Rule).
Define the coefficients of the equations:
For the first equation [tex]\(2x + 5y = 31\)[/tex]:
- [tex]\(a_1 = 2\)[/tex]
- [tex]\(b_1 = 5\)[/tex]
- [tex]\(c_1 = 31\)[/tex]
For the second equation [tex]\(3x - 2y = -1\)[/tex]:
- [tex]\(a_2 = 3\)[/tex]
- [tex]\(b_2 = -2\)[/tex]
- [tex]\(c_2 = -1\)[/tex]
1. Calculate the determinant of the coefficient matrix [tex]\( D \)[/tex]:
[tex]\[
D = a_1b_2 - a_2b_1 = (2)(-2) - (3)(5) = -4 - 15 = -19
\][/tex]
2. Calculate the determinant for [tex]\( x \)[/tex] ( [tex]\( D_x \)[/tex] ):
[tex]\[
D_x = c_1b_2 - c_2b_1 = (31)(-2) - (-1)(5) = -62 + 5 = -57
\][/tex]
3. Calculate the determinant for [tex]\( y \)[/tex] ( [tex]\( D_y \)[/tex] ):
[tex]\[
D_y = a_1c_2 - a_2c_1 = (2)(-1) - (3)(31) = -2 - 93 = -95
\][/tex]
4. Now, find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
x = \frac{D_x}{D} = \frac{-57}{-19} = 3
\][/tex]
[tex]\[
y = \frac{D_y}{D} = \frac{-95}{-19} = 5
\][/tex]
5. Finally, calculate [tex]\( x - y \)[/tex]:
[tex]\[
x - y = 3 - 5 = -2
\][/tex]
Thus, the value of [tex]\( x - y \)[/tex] is [tex]\(-2\)[/tex].
