Bapsaur 76 (2024 r.) 1) ) x_(1)-2x_(2)+2x_(3)-4x_(4)=-2 -5x_(1)+8x_(2)-4x_(3)+12x_(4)=-4 x_(1)-7x_(2)+5x_(3)-12x_(4)=-1 -2x_ 2) ) x_(1)+x_(2)+2x_(3)=1 2x_(1)-x_(2)+2x_(3)=-4 4x_(1)+x_(2)+4x_(3)=-2

1) To solve the system of equations, we can use the method of substitution or elimination. Let's use the elimination method.<br /><br />First, let's multiply the first equation by 2 and the second equation by 1 to make the coefficients of $x_1$ in both equations equal:<br /><br />$\begin{cases} 2x_{1}-4x_{2}+4x_{3}-8x_{4}=-4\\ -5x_{1}+8x_{2}-4x_{3}+12x_{4}=-4\end{cases}$<br /><br />Now, let's add the two equations:<br /><br />$(2x_{1}-4x_{2}+4x_{3}-8x_{4})+(-5x_{1}+8x_{2}-4x_{3}+12x_{4})=-4+(-4)$<br /><br />Simplifying the equation:<br /><br />$-3x_{1}+4x_{2}+8x_{3}+4x_{4}=-8$<br /><br />Now, let's multiply the third equation by 1 and the fourth equation by 2 to make the coefficients of $x_1$ in both equations equal:<br /><br />$\begin{cases} x_{1}-7x_{2}+5x_{3}-12x_{4}=-1\\ -2x_{1}+4x_{2}-8x_{3}+16x_{4}=-2\end{cases}$<br /><br />Now, let's add the two equations:<br /><br />$(x_{1}-7x_{2}+5x_{3}-12x_{4})+(-2x_{1}+4x_{2}-8x_{3}+16x_{4})=-1+(-2)$<br /><br />Simplifying the equation:<br /><br />$-x_{1}-3x_{2}-3x_{3}+4x_{4}=-3$<br /><br />Now we have two new equations:<br /><br />$\begin{cases} -3x_{1}+4x_{2}+8x_{3}+4x_{4}=-8\\ -x_{1}-3x_{2}-3x_{3}+4x_{4}=-3\end{cases}$<br /><br />We can solve this system of equations using the elimination method again. Let's multiply the second equation by 3 to make the coefficients of $x_1$ in both equations equal:<br /><br />$\begin{cases} -3x_{1}+4x_{2}+8x_{3}+4x_{4}=-8\\ -3x_{1}-9x_{2}-9x_{3}+12x_{4}=-9\end{cases}$<br /><br />Now, let's subtract the second equation from the first:<br /><br />$(-3x_{1}+4x_{2}+8x_{3}+4x_{4})-(-3x_{1}-9x_{2}-9x_{3}+12x_{4})=-8-(-9)$<br /><br />Simplifying the equation:<br /><br />$13x_{2}+17x_{3}-8x_{4}=1$<br /><br />Now we have a new equation:<br /><br />$13x_{2}+17x_{3}-8x_{4}=1$<br /><br />We can solve this equation using the substitution method. Let's solve for $x_2$:<br /><br />$x_{2}=\frac{1-17x_{3}+8x_{4}}{13}$<br /><br />Now, let's substitute this value of $x_2$ into the first equation:<br /><br />$-3x_{1}+4\left(\frac{1-17x_{3}+8x_{4}}{13}\right)+8x_{3}+4x_{4}=-8$<br /><br />Simplifying the equation:<br /><br />$-3x_{1}+\frac{4-68x_{3}+32x_{4}}{13}+8x_{3}+4x_{4}=-8$<br /><br />Multiplying through by 13 to clear the fraction:<br /><br />$-39x_{1}+4-68x_{3}+32x_{4}+104x_{3}+52x_{4}=-104$<br /><br />Combining like terms:<br /><br />$65x_{3}+84x_{4}=-108$<br /><br />Now, let's solve for $x_3$:<br /><br />$x_{3}=\frac{-108-84x_{4}}{65}$<br /><br />Now, let's substitute this value of $x_3$ into the equation for $x_2$:<br /><br />$x_{2}=\frac{1-17\left(\frac{-108-84x_{4}}{65}\right)+8x_{4}}{13}$<br /><br />Simplifying the equation:<br /><br />$x_{2}=\frac{1+\frac{1836+1428x_{4}}{65