1. Determine $\Delta = a_{11}a_{22} - a_{12}a_{21}$ for system \(\begin{cases} x + y = 3 \\ x + 2y = -8 \end{cases}\). Coefficients: $a_{11}=1$, $a_{12}=1$, $a_{21}=1$, $a_{22}=2$ Calculate $\Delta = 1 \times 2 - 1 \times 1 = 2 - 1 = 1$. Solve: From first: $x = 3 - y$ Plug into second: $$ 3 - y + 2y = -8 \\ 3 + y = -8 \\ y = -11 $$ Calculate $x$: $$ x = 3 - (-11) = 14 $$ Solution: $x = 14$, $y = -11$, $\Delta = 1$. 2. For system \(\begin{cases} 2x + 3y = 3 \\ -2x - 3y = -3 \end{cases}\), Coefficients: $a_{11}=2$, $a_{12}=3$, $a_{21}=-2$, $a_{22}=-3$ Calculate $\Delta = 2 \times (-3) - 3 \times (-2) = -6 + 6 = 0$. Since $\Delta=0$, system may have infinite or no solutions. Add equations: $$ (2x - 2x) + (3y -3y) = 3 -3 \\ 0=0 $$ Equations are dependent, infinite solutions along line $2x + 3y = 3$. 3. System \(\begin{cases} 4x + 5y = 0 \\ -2x - y = 3 \end{cases}\). Coefficients: $a_{11}=4$, $a_{12}=5$, $a_{21}=-2$, $a_{22}=-1$ Calculate $\Delta = 4 \times (-1) - 5 \times (-2) = -4 + 10 = 6$. Non-zero determinant implies unique solution. Solve: From second:\ $$ -2x - y = 3 \\ y = -2x - 3 $$ Substitute into first: $$ 4x + 5(-2x -3) = 0 \\ 4x - 10x - 15 = 0 \\ -6x = 15 \\ x = -\frac{15}{6} = -\frac{5}{2} $$ Calculate $y$: $$ y = -2(-\frac{5}{2}) - 3 = 5 - 3 = 2 $$ Solution: $x = -\frac{5}{2}$, $y=2$, $\Delta=6$. 4. System \(\begin{cases} -2x = 1 \\ 4x - 3y = 0 \end{cases}\). Rewrite first: $-2x=1 \Rightarrow x = -\frac{1}{2}$ Coefficients for $\Delta$: $a_{11}= -2$, $a_{12}=0$ (no $y$ term), $a_{21}=4$, $a_{22}=-3$ Calculate $\Delta = (-2)(-3) - 0 \times 4 = 6$ Substitute $x$ in second: $$ 4(-\frac{1}{2}) - 3y = 0 \\ -2 - 3y = 0 \\ -3y = 2 \\ y = -\frac{2}{3} $$ Solution: $x = -\frac{1}{2}$, $y = -\frac{2}{3}$, $\Delta=6$. 5. System \(\begin{cases} 7x + 3y = 0 \\ -5x + 10y = 0 \end{cases}\). Coefficients: $a_{11}=7$, $a_{12}=3$, $a_{21}=-5$, $a_{22}=10$ Calculate $\Delta = 7 imes 10 - 3 imes (-5) = 70 + 15 = 85$ Unique solution. Solve by elimination or substitution: Multiply first by 5: $$ 35x + 15y = 0 $$ Multiply second by 7: $$ -35x + 70y = 0 $$ Add: $$ 0x + 85y = 0 \Rightarrow y = 0 $$ From first: $$ 7x + 3(0) = 0 \Rightarrow x=0 $$ Solution: $x=0$, $y=0$, $\Delta=85$. 6. System \(\begin{cases} 3x - 7y = -5 \\ 4x - 3y = -2 \end{cases}\). Coefficients: $a_{11}=3$, $a_{12}=-7$, $a_{21}=4$, $a_{22}=-3$ Calculate $\Delta = 3 imes (-3) - (-7) imes 4 = -9 + 28 = 19$ Solve: Multiply first by 3: $$ 9x - 21y = -15 $$ Multiply second by 7: $$ 28x - 21y = -14 $$ Subtract: $$ (28x - 9x) - (21y - 21y) = -14 + 15 \\ 19x = 1 \\ x = \frac{1}{19} $$ Calculate $y$: $$ 3(\frac{1}{19}) - 7y = -5 \\ \frac{3}{19} - 7y = -5 \\ -7y = -5 - \frac{3}{19} = -\frac{95}{19} - \frac{3}{19} = -\frac{98}{19} \\ y = \frac{98}{133} $$ Solution: $x=\frac{1}{19}$, $y=\frac{98}{133}$, $\Delta=19$. 7. System \(\begin{cases} 7x + 4y = 1 \\ -7x - 4y = -3 \end{cases}\). Calculate $\Delta = 7 imes (-4) - 4 imes (-7) = -28 + 28 = 0$. Add equations: $$ (7x -7x) + (4y -4y) = 1 -3 \\ 0 = -2 $$ Contradiction implies no solution. $\Delta=0$, no solution. 8. System \(\begin{cases} 7x + 4y = 0 \\ -7x - 4y = 0 \end{cases}\). $\Delta=0$ (same as previous). Add: $$ 0=0 $$ Dependent system, infinite solutions along $7x + 4y = 0$. 9. System \(\begin{cases} -13x + 3y = 7 \\ 5x + 22y = 9 \end{cases}\). Coefficients: $a_{11}=-13$, $a_{12}=3$, $a_{21}=5$, $a_{22}=22$ Calculate $\Delta = (-13)(22) - 3 imes 5 = -286 -15 = -301$ Solve: Multiply first by 5: $$ -65x + 15y = 35 $$ Multiply second by 13: $$ 65x + 286y = 117 $$ Add: $$ 0x + 301y = 152 \\ y = \frac{152}{301} $$ Calculate $x$: $$ -13x + 3(\frac{152}{301}) = 7 \\ -13x + \frac{456}{301} = 7 \\ -13x = 7 - \frac{456}{301} = \frac{2107}{301} - \frac{456}{301} = \frac{1651}{301} \\ x = -\frac{1651}{301 \times 13} = -\frac{1651}{3913} $$ Solution: $x = -\frac{1651}{3913}$, $y = \frac{152}{301}$, $\Delta = -301$. 10. System \(\begin{cases} 9x - 3y = -3 \\ -2x + 4y = 1 \end{cases}\). $\Delta = 9 imes 4 - (-3)(-2) = 36 - 6 = 30$ Solve: Multiply first by 4: $$ 36x - 12y = -12 $$ Multiply second by 3: $$ -6x + 12y = 3 $$ Add: $$ 30x = -9 \\ x = -\frac{3}{10} $$ Calculate $y$: $$ 9(-\frac{3}{10}) - 3y = -3 \\ -\frac{27}{10} - 3y = -3 \\ -3y = -3 + \frac{27}{10} = -\frac{30}{10} + \frac{27}{10} = -\frac{3}{10} \\ y = \frac{1}{10} $$ Solution: $x = -\frac{3}{10}$, $y = \frac{1}{10}$, $\Delta = 30$. 11. System \(\begin{cases} -2x + 3y = 3 \\ 2x - 3y = -3 \end{cases}\). $\Delta = (-2)(-3) - 3 imes 2 = 6 - 6 = 0$ Add equations: $$ 0=0 $$ Dependent system, infinite solutions on line $-2x + 3y= 3$. 12. System \(\begin{cases} x + 2y = 5 \\ 3x + 4y = 6 \end{cases}\). $\Delta = 1 imes 4 - 2 imes 3 = 4 - 6 = -2$ Solve: From first: $$ x = 5 - 2y $$ Substitute in second: $$ 3(5 - 2y) + 4y = 6 \\ 15 - 6y + 4y = 6 \\ -2y = -9 \\ y = \frac{9}{2} = 4.5 $$ Calculate $x$: $$ x = 5 - 2(4.5) = 5 - 9 = -4 $$ Solution: $x = -4$, $y = \frac{9}{2}$, $\Delta = -2$. 13. System \(\begin{cases} y = -3 \\ -2x + 4y = 8 \end{cases}\). $\Delta$ is not defined because no coefficient of $x$ and $y$ for first equation. Substitute $y$ in second: $$ -2x + 4(-3) = 8 \\ -2x -12 = 8 \\ -2x = 20 \\ x = -10 $$ Solution: $x = -10$, $y = -3$, determinant not applicable. 14. System \(\begin{cases} -7x + 2y = -9 \\ 2y = -6 \end{cases}\). From second: $2y = -6 \Rightarrow y = -3$ Calculate $\Delta = -7 imes 0 - 2 imes 0 = 0$ because second equation lacks $x$ term (consider coefficients as $a_{21}=0$, $a_{22}=2$). Substitute $y$ into first: $$ -7x + 2(-3) = -9 \\ -7x - 6 = -9 \\ -7x = -3 \\ x = \frac{3}{7} $$ Solution: $x=\frac{3}{7}$, $y=-3$, $\Delta = -14$, correcting coefficients to $a_{22}=2$, $\Delta = (-7)(2)-2(0)= -14$ actual coefficients. Final: $\Delta$ values calculated for all systems; solutions found when $\Delta \neq 0$; infinite or no solutions when $\Delta=0$.