Inicio › Matemáticas 3 › Método de igualación

Matemáticas 3 · Tema 4

Método de igualación

Ejemplo

{2x−3y=122x−8y=22\begin{cases} 2x - 3y = 12 \\ 2x - 8y = 22 \end{cases}
Despejamos e igualamos
2x=12+3y2x=22+8y\begin{aligned} 2x &= 12 + 3y \\ 2x &= 22 + 8y \end{aligned} 12+3y=22+8y12 + 3y = 22 + 8y 3y−8y=22−12−5y=10y=10−5\begin{aligned} 3y - 8y &= 22 - 12 \\ -5y &= 10 \\ y &= \dfrac{10}{-5} \end{aligned} y=−2y = -2
Sustituimos
2x−3(−2)=122x+6=122x=12−62x=6x=62\begin{aligned} 2x - 3(-2) &= 12 \\ 2x + 6 &= 12 \\ 2x &= 12 - 6 \\ 2x &= 6 \\ x &= \dfrac{6}{2} \end{aligned} x=3x = 3

Comprobación

2(3)−3(−2)=126+6=1212=12\begin{aligned} 2(3) - 3(-2) &= 12 \\ 6 + 6 &= 12 \\ 12 &= 12 \end{aligned}
2(3)−8(−2)=226+16=2222=22\begin{aligned} 2(3) - 8(-2) &= 22 \\ 6 + 16 &= 22 \\ 22 &= 22 \end{aligned}

Ejercicios

Ejercicio 1
{3x−2y=93x+4y=−9\begin{cases} 3x - 2y = 9 \\ 3x + 4y = -9 \end{cases}
Ejercicio 2
{6x−2y=26x+3y=12\begin{cases} 6x - 2y = 2 \\ 6x + 3y = 12 \end{cases}
Ejercicio 3
{x+2y=44x+2y=−2\begin{cases} x + 2y = 4 \\ 4x + 2y = -2 \end{cases}
Ejercicio 4
{−2x+3y=−8x+3y=−5\begin{cases} -2x + 3y = -8 \\ x + 3y = -5 \end{cases}

Respuestas

▼
Ejercicio 1
{3x−2y=93x+4y=−9\begin{cases} 3x - 2y = 9 \\ 3x + 4y = -9 \end{cases}
Despejamos e igualamos
3x=9+2y3x=−9−4y\begin{aligned} 3x &= 9 + 2y \\ 3x &= -9 - 4y \end{aligned} −9−4y=9+2y-9 - 4y = 9 + 2y −4y−2y=9+9−6y=18y=18−6\begin{aligned} -4y - 2y &= 9 + 9 \\ -6y &= 18 \\ y &= \dfrac{18}{-6} \end{aligned} y=−3y = -3
Sustituimos
3x−2(−3)=93x+6=93x=9−63x=3x=33\begin{aligned} 3x - 2(-3) &= 9 \\ 3x + 6 &= 9 \\ 3x &= 9 - 6 \\ 3x &= 3 \\ x &= \dfrac{3}{3} \end{aligned} x=1x = 1

Comprobación

3(1)−2(−3)=93+6=99=9\begin{aligned} 3(1) - 2(-3) &= 9 \\ 3 + 6 &= 9 \\ 9 &= 9 \end{aligned}
3(1)+4(−3)=−93−12=−9−9=−9\begin{aligned} 3(1) + 4(-3) &= -9 \\ 3 - 12 &= -9 \\ -9 &= -9 \end{aligned}
Ejercicio 2
{6x−2y=26x+3y=12\begin{cases} 6x - 2y = 2 \\ 6x + 3y = 12 \end{cases}
Despejamos e igualamos
6x=2+2y6x=12−3y\begin{aligned} 6x &= 2 + 2y \\ 6x &= 12 - 3y \end{aligned} 12−3y=2+2y12 - 3y = 2 + 2y −3y−2y=2−12−5y=−10y=−10−5\begin{aligned} -3y - 2y &= 2 - 12 \\ -5y &= -10 \\ y &= \dfrac{-10}{-5} \end{aligned} y=2y = 2
Sustituimos
6x−2(2)=26x−4=26x=2+46x=6x=66\begin{aligned} 6x - 2(2) &= 2 \\ 6x - 4 &= 2 \\ 6x &= 2 + 4 \\ 6x &= 6 \\ x &= \dfrac{6}{6} \end{aligned} x=1x = 1

Comprobación

6(1)−2(2)=26−4=22=2\begin{aligned} 6(1) - 2(2) &= 2 \\ 6 - 4 &= 2 \\ 2 &= 2 \end{aligned}
6(1)+3(2)=126+6=1212=12\begin{aligned} 6(1) + 3(2) &= 12 \\ 6 + 6 &= 12 \\ 12 &= 12 \end{aligned}
Ejercicio 3
{x+2y=44x+2y=−2\begin{cases} x + 2y = 4 \\ 4x + 2y = -2 \end{cases}
Despejamos e igualamos
2y=4−x2y=−2−4x\begin{aligned} 2y &= 4 - x \\ 2y &= -2 - 4x \end{aligned} −2−4x=4−x-2 - 4x = 4 - x −4x+x=4+2−3x=6x=6−3\begin{aligned} -4x + x &= 4 + 2 \\ -3x &= 6 \\ x &= \dfrac{6}{-3} \end{aligned} x=−2x = -2
Sustituimos
−2+2y=42y=4+22y=6y=62\begin{aligned} -2 + 2y &= 4 \\ 2y &= 4 + 2 \\ 2y &= 6 \\ y &= \dfrac{6}{2} \end{aligned} y=3y = 3

Comprobación

(−2)+2(3)=4−2+6=44=4\begin{aligned} (-2) + 2(3) &= 4 \\ -2 + 6 &= 4 \\ 4 &= 4 \end{aligned}
4(−2)+2(3)=−2−8+6=−2−2=−2\begin{aligned} 4(-2) + 2(3) &= -2 \\ -8 + 6 &= -2 \\ -2 &= -2 \end{aligned}
Ejercicio 4
{−2x+3y=−8x+3y=−5\begin{cases} -2x + 3y = -8 \\ x + 3y = -5 \end{cases}
Despejamos e igualamos
3y=−8+2x3y=−5−x\begin{aligned} 3y &= -8 + 2x \\ 3y &= -5 - x \end{aligned} −5−x=−8+2x-5 - x = -8 + 2x −x−2x=−8+5−3x=−3x=−3−3\begin{aligned} -x - 2x &= -8 + 5 \\ -3x &= -3 \\ x &= \dfrac{-3}{-3} \end{aligned} x=1x = 1
Sustituimos
−2(1)+3y=−8−2+3y=−83y=−8+23y=−6y=−63\begin{aligned} -2(1) + 3y &= -8 \\ -2 + 3y &= -8 \\ 3y &= -8 + 2 \\ 3y &= -6 \\ y &= \dfrac{-6}{3} \end{aligned} y=−2y = -2

Comprobación

−2(1)+3(−2)=−8−2−6=−8−8=−8\begin{aligned} -2(1) + 3(-2) &= -8 \\ -2 - 6 &= -8 \\ -8 &= -8 \end{aligned}
(1)+3(−2)=−51−6=−5−5=−5\begin{aligned} (1) + 3(-2) &= -5 \\ 1 - 6 &= -5 \\ -5 &= -5 \end{aligned}

Ejercicios extra

▼
Bloque 1
{4x−3y=−14x+2y=14\begin{cases} 4x - 3y = -1 \\ 4x + 2y = 14 \end{cases}
{2x+5y=12x−3y=9\begin{cases} 2x + 5y = 1 \\ 2x - 3y = 9 \end{cases}
{5x+3y=2−x+3y=14\begin{cases} 5x + 3y = 2 \\ -x + 3y = 14 \end{cases}
{3x−2y=−73x+4y=23\begin{cases} 3x - 2y = -7 \\ 3x + 4y = 23 \end{cases}
{4x+2y=−16−x+2y=−1\begin{cases} 4x + 2y = -16 \\ -x + 2y = -1 \end{cases}
Bloque 2
{5x+3y=235x−2y=18\begin{cases} 5x + 3y = 23 \\ 5x - 2y = 18 \end{cases}
{2x−4y=−263x−4y=−27\begin{cases} 2x - 4y = -26 \\ 3x - 4y = -27 \end{cases}
{2x+7y=−42x−3y=16\begin{cases} 2x + 7y = -4 \\ 2x - 3y = 16 \end{cases}
{3x+5y=3−2x+5y=23\begin{cases} 3x + 5y = 3 \\ -2x + 5y = 23 \end{cases}
{6x+y=76x−4y=32\begin{cases} 6x + y = 7 \\ 6x - 4y = 32 \end{cases}
← Método gráfico Determinantes 2×2 →