Ejercicios Resueltos de Sistema de Ecuaciones

a)  $\left \{ \begin{matrix} 5x-2=y
\\ y=2x-1 \end{matrix}\right.$ → $\left \{ \begin{matrix} 5x-2=y
\\ 2x-1=y \end{matrix}\right.$ → $\left \{ \begin{matrix} 10x-4=2y
\\ 10x-5=5y \end{matrix}\right.$
 → $1=-3y$ → $y=-1/3\bullet$ → $x=1/3\bullet$

b)  $\left \{ \begin{matrix} x/5-y/3=16
\\ x/2+y/5=9 \end{matrix}\right.$ → $\left \{ \begin{matrix} x/10-y/6=8
\\ x/10+y/25=9/5 \end{matrix}\right.$
 → $-y/6-y/25=8-9/5$  → $-y/6-y/25=31/5$
→ $-25y-6y=930$ → $-31y=930$ → $y=-930/31$
→ $y=-30\bullet$ → $x=30\bullet$

c)  $\left \{ \begin{matrix} \displaystyle\frac{x+y}{2}+\displaystyle\frac{x-y}{3}=\displaystyle\frac{17}{6}
\\ \displaystyle\frac{x+y}{3}-\displaystyle\frac{x-y}{2}=\displaystyle\frac{11}{2} \end{matrix}\right.$ → $\left \{ \begin{matrix} \displaystyle\frac{3x+3y}{6}+\displaystyle\frac{2x-2y}{6}=\displaystyle\frac{17}{6}
\\ \displaystyle\frac{2x+2y}{6}-\displaystyle\frac{3x-3y}{6}=\displaystyle\frac{33}{6} \end{matrix}\right.$
 → $(3x+3y)+(2x-2y)=17$  → $5x+y=17$
→ $(2x+2y)-(3x-3y)=33$  → $-x+5y=33$
→ $\left \{ \begin{matrix} 5x+y=17
\\ -x+5y=33 \end{matrix}\right.$ → $\left \{ \begin{matrix} 5x+y=17
\\ -5x+25y=165 \end{matrix}\right.$
→ $26y=182$ → $y=182/26$ → $y=7\bullet$ → $x=2\bullet$

d)  $\left \{ \begin{matrix} \displaystyle\frac{3x-2y+1}{6}-\displaystyle\frac{5x-2y+5}{2}=-1
\\ (x+1)^2-3y=(x-2)^2 \end{matrix}\right.$
→ $\left \{ \begin{matrix} 3x-2y+1-3(5x-2y+5)=-6
\\ x^2+2x+1-3y=x^2-4x+4 \end{matrix}\right.$
→ $\left \{ \begin{matrix} -12x+4y=8
\\ 6x-3y=3 \end{matrix}\right.$ → $\left \{ \begin{matrix} -12x+4y=8
\\ 12x-6y=6 \end{matrix}\right.$
→ $-2y=14$ → $y=-7\bullet$ → $x=-3\bullet$