Inicio › Matemáticas 3 › Método de sustitución

Matemáticas 3 · Tema 2

Método de sustitución

Video

Ejemplo

{3x+y=105x−2y=2⟶y=10−3x\begin{cases} 3x + y = 10 \\ 5x - 2y = 2 \end{cases} \longrightarrow \color{#7B2CBF}{y = 10 - 3x}
Sustituimos
5x−2(10−3x)=25x−20+6x=211x=2+2011x=22x=2211\begin{aligned} 5x - 2(10 - 3x) &= 2 \\ 5x - 20 + 6x &= 2 \\ 11x &= 2 + 20 \\ 11x &= 22 \\ x &= \dfrac{22}{11} \end{aligned} x=2x = 2
Regresamos al despeje
y=10−3(2)y=10−6\begin{aligned} y &= 10 - 3(2) \\ y &= 10 - 6 \end{aligned} y=4y = 4

Comprobación

3(2)+(4)=106+4=1010=10\begin{aligned} 3(2) + (4) &= 10 \\ 6 + 4 &= 10 \\ 10 &= 10 \end{aligned}
5(2)−2(4)=210−8=22=2\begin{aligned} 5(2) - 2(4) &= 2 \\ 10 - 8 &= 2 \\ 2 &= 2 \end{aligned}

Ejercicios

Ejercicio 1
{5x−2y=17x+4y=−1\begin{cases} 5x - 2y = 17 \\ x + 4y = -1 \end{cases}
Ejercicio 2
{3x+y=−12x−3y=−19\begin{cases} 3x + y = -1 \\ 2x - 3y = -19 \end{cases}
Ejercicio 3
{2x+3y=−65x+y=11\begin{cases} 2x + 3y = -6 \\ 5x + y = 11 \end{cases}
Ejercicio 4
{x+2y=93x−y=−15\begin{cases} x + 2y = 9 \\ 3x - y = -15 \end{cases}

Respuestas

▼
Ejercicio 1
{5x−2y=17x+4y=−1⟶x=−1−4y\begin{cases} 5x - 2y = 17 \\ x + 4y = -1 \end{cases} \longrightarrow \color{#7B2CBF}{x = -1 - 4y}
Sustituimos
5(−1−4y)−2y=17−5−20y−2y=17−22y=17+5−22y=22y=22−22\begin{aligned} 5(-1 - 4y) - 2y &= 17 \\ -5 - 20y - 2y &= 17 \\ -22y &= 17 + 5 \\ -22y &= 22 \\ y &= \dfrac{22}{-22} \end{aligned} y=−1y = -1
Regresamos al despeje
x=−1−4(−1)x=−1+4\begin{aligned} x &= -1 - 4(-1) \\ x &= -1 + 4 \end{aligned} x=3x = 3

Comprobación

5(3)−2(−1)=1715+2=1717=17\begin{aligned} 5(3) - 2(-1) &= 17 \\ 15 + 2 &= 17 \\ 17 &= 17 \end{aligned}
(3)+4(−1)=−13−4=−1−1=−1\begin{aligned} (3) + 4(-1) &= -1 \\ 3 - 4 &= -1 \\ -1 &= -1 \end{aligned}
Ejercicio 2
{3x+y=−12x−3y=−19⟶y=−1−3x\begin{cases} 3x + y = -1 \\ 2x - 3y = -19 \end{cases} \longrightarrow \color{#7B2CBF}{y = -1 - 3x}
Sustituimos
2x−3(−1−3x)=−192x+3+9x=−1911x=−19−311x=−22x=−2211\begin{aligned} 2x - 3(-1 - 3x) &= -19 \\ 2x + 3 + 9x &= -19 \\ 11x &= -19 - 3 \\ 11x &= -22 \\ x &= \dfrac{-22}{11} \end{aligned} x=−2x = -2
Regresamos al despeje
y=−1−3(−2)y=−1+6\begin{aligned} y &= -1 - 3(-2) \\ y &= -1 + 6 \end{aligned} y=5y = 5

Comprobación

3(−2)+(5)=−1−6+5=−1−1=−1\begin{aligned} 3(-2) + (5) &= -1 \\ -6 + 5 &= -1 \\ -1 &= -1 \end{aligned}
2(−2)−3(5)=−19−4−15=−19−19=−19\begin{aligned} 2(-2) - 3(5) &= -19 \\ -4 - 15 &= -19 \\ -19 &= -19 \end{aligned}
Ejercicio 3
{2x+3y=−65x+y=11⟶y=11−5x\begin{cases} 2x + 3y = -6 \\ 5x + y = 11 \end{cases} \longrightarrow \color{#7B2CBF}{y = 11 - 5x}
Sustituimos
2x+3(11−5x)=−62x+33−15x=−6−13x=−6−33−13x=−39x=−39−13\begin{aligned} 2x + 3(11 - 5x) &= -6 \\ 2x + 33 - 15x &= -6 \\ -13x &= -6 - 33 \\ -13x &= -39 \\ x &= \dfrac{-39}{-13} \end{aligned} x=3x = 3
Regresamos al despeje
y=11−5(3)y=11−15\begin{aligned} y &= 11 - 5(3) \\ y &= 11 - 15 \end{aligned} y=−4y = -4

Comprobación

2(3)+3(−4)=−66−12=−6−6=−6\begin{aligned} 2(3) + 3(-4) &= -6 \\ 6 - 12 &= -6 \\ -6 &= -6 \end{aligned}
5(3)+(−4)=1115−4=1111=11\begin{aligned} 5(3) + (-4) &= 11 \\ 15 - 4 &= 11 \\ 11 &= 11 \end{aligned}
Ejercicio 4
{x+2y=93x−y=−15⟶x=9−2y\begin{cases} x + 2y = 9 \\ 3x - y = -15 \end{cases} \longrightarrow \color{#7B2CBF}{x = 9 - 2y}
Sustituimos
3(9−2y)−y=−1527−6y−y=−15−7y=−15−27−7y=−42y=−42−7\begin{aligned} 3(9 - 2y) - y &= -15 \\ 27 - 6y - y &= -15 \\ -7y &= -15 - 27 \\ -7y &= -42 \\ y &= \dfrac{-42}{-7} \end{aligned} y=6y = 6
Regresamos al despeje
x=9−2(6)x=9−12\begin{aligned} x &= 9 - 2(6) \\ x &= 9 - 12 \end{aligned} x=−3x = -3

Comprobación

(−3)+2(6)=9−3+12=99=9\begin{aligned} (-3) + 2(6) &= 9 \\ -3 + 12 &= 9 \\ 9 &= 9 \end{aligned}
3(−3)−(6)=−15−9−6=−15−15=−15\begin{aligned} 3(-3) - (6) &= -15 \\ -9 - 6 &= -15 \\ -15 &= -15 \end{aligned}

Ejercicios extra

▼
Bloque 1
{x+3y=72x−5y=3\begin{cases} x + 3y = 7 \\ 2x - 5y = 3 \end{cases}
{4x+y=93x−2y=4\begin{cases} 4x + y = 9 \\ 3x - 2y = 4 \end{cases}
{2x+y=−35x−3y=20\begin{cases} 2x + y = -3 \\ 5x - 3y = 20 \end{cases}
{x−4y=−143x+2y=0\begin{cases} x - 4y = -14 \\ 3x + 2y = 0 \end{cases}
{3x+y=52x−3y=7\begin{cases} 3x + y = 5 \\ 2x - 3y = 7 \end{cases}
Bloque 2
{x+5y=114x−3y=−2\begin{cases} x + 5y = 11 \\ 4x - 3y = -2 \end{cases}
{2x+y=83x+4y=17\begin{cases} 2x + y = 8 \\ 3x + 4y = 17 \end{cases}
{x−2y=−15x+3y=21\begin{cases} x - 2y = -1 \\ 5x + 3y = 21 \end{cases}
{6x−y=132x+5y=−1\begin{cases} 6x - y = 13 \\ 2x + 5y = -1 \end{cases}
{x+7y=43x−2y=−11\begin{cases} x + 7y = 4 \\ 3x - 2y = -11 \end{cases}
← Método de sumas y restas Método gráfico →