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Matemáticas 3 · Tema 1

Método de sumas y restas

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Ejemplo

(−5)(2){2x+4y=−85x−3y=19}(3)(4)\begin{array}{ccc} \begin{array}{r} \color{#7B2CBF}{(-5)} \\ \color{#7B2CBF}{(2)} \end{array} & \left\{\begin{array}{l} 2x + 4y = -8 \\ 5x - 3y = 19 \end{array}\right\} & \begin{array}{l} \color{#FF00AA}{(3)} \\ \color{#FF00AA}{(4)} \end{array} \end{array}
Eliminamos xx
−10x−20y=4010x−6y=38−26y=78\begin{array}{r} -10x - 20y = 40 \\ 10x - 6y = 38 \\ \hline -26y = 78 \end{array} y=78−26y = \dfrac{78}{-26} y=−3y = -3
Eliminamos yy
6x+12y=−2420x−12y=7626x=52\begin{array}{r} 6x + 12y = -24 \\ 20x - 12y = 76 \\ \hline 26x = 52 \end{array} x=5226x = \dfrac{52}{26} x=2x = 2

Comprobación

2(2)+4(−3)=−84−12=−8−8=−8\begin{aligned} 2(2) + 4(-3) &= -8 \\ 4 - 12 &= -8 \\ -8 &= -8 \end{aligned}
5(2)−3(−3)=1910+9=1919=19\begin{aligned} 5(2) - 3(-3) &= 19 \\ 10 + 9 &= 19 \\ 19 &= 19 \end{aligned}

Ejercicios

Ejercicio 1
{2x+5y=−33x−4y=30\begin{cases} 2x + 5y = -3 \\ 3x - 4y = 30 \end{cases}
Ejercicio 2
{3x−4y=−322x+2y=2\begin{cases} 3x - 4y = -32 \\ 2x + 2y = 2 \end{cases}
Ejercicio 3
{3x+4y=17x−6y=33\begin{cases} 3x + 4y = 1 \\ 7x - 6y = 33 \end{cases}
Ejercicio 4
{4x−2y=22x−8y=13\begin{cases} 4x - 2y = 22 \\ x - 8y = 13 \end{cases}

Respuestas

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Ejercicio 1
(−3)(2){2x+5y=−33x−4y=30}(4)(5)\begin{array}{ccc} \begin{array}{r} \color{#7B2CBF}{(-3)} \\ \color{#7B2CBF}{(2)} \end{array} & \left\{\begin{array}{l} 2x + 5y = -3 \\ 3x - 4y = 30 \end{array}\right\} & \begin{array}{l} \color{#FF00AA}{(4)} \\ \color{#FF00AA}{(5)} \end{array} \end{array}
Eliminamos xx
−6x−15y=96x−8y=60−23y=69\begin{array}{r} -6x - 15y = 9 \\ 6x - 8y = 60 \\ \hline -23y = 69 \end{array} y=69−23y = \dfrac{69}{-23} y=−3y = -3
Eliminamos yy
8x+20y=−1215x−20y=15023x=138\begin{array}{r} 8x + 20y = -12 \\ 15x - 20y = 150 \\ \hline 23x = 138 \end{array} x=13823x = \dfrac{138}{23} x=6x = 6

Comprobación

2(6)+5(−3)=−312−15=−3−3=−3\begin{aligned} 2(6) + 5(-3) &= -3 \\ 12 - 15 &= -3 \\ -3 &= -3 \end{aligned}
3(6)−4(−3)=3018+12=3030=30\begin{aligned} 3(6) - 4(-3) &= 30 \\ 18 + 12 &= 30 \\ 30 &= 30 \end{aligned}
Ejercicio 2
(−2)(3){3x−4y=−322x+2y=2}(2)(4)\begin{array}{ccc} \begin{array}{r} \color{#7B2CBF}{(-2)} \\ \color{#7B2CBF}{(3)} \end{array} & \left\{\begin{array}{l} 3x - 4y = -32 \\ 2x + 2y = 2 \end{array}\right\} & \begin{array}{l} \color{#FF00AA}{(2)} \\ \color{#FF00AA}{(4)} \end{array} \end{array}
Eliminamos xx
−6x+8y=646x+6y=614y=70\begin{array}{r} -6x + 8y = 64 \\ 6x + 6y = 6 \\ \hline 14y = 70 \end{array} y=7014y = \dfrac{70}{14} y=5y = 5
Eliminamos yy
6x−8y=−648x+8y=814x=−56\begin{array}{r} 6x - 8y = -64 \\ 8x + 8y = 8 \\ \hline 14x = -56 \end{array} x=−5614x = \dfrac{-56}{14} x=−4x = -4

Comprobación

3(−4)−4(5)=−32−12−20=−32−32=−32\begin{aligned} 3(-4) - 4(5) &= -32 \\ -12 - 20 &= -32 \\ -32 &= -32 \end{aligned}
2(−4)+2(5)=2−8+10=22=2\begin{aligned} 2(-4) + 2(5) &= 2 \\ -8 + 10 &= 2 \\ 2 &= 2 \end{aligned}
Ejercicio 3
(−7)(3){3x+4y=17x−6y=33}(6)(4)\begin{array}{ccc} \begin{array}{r} \color{#7B2CBF}{(-7)} \\ \color{#7B2CBF}{(3)} \end{array} & \left\{\begin{array}{l} 3x + 4y = 1 \\ 7x - 6y = 33 \end{array}\right\} & \begin{array}{l} \color{#FF00AA}{(6)} \\ \color{#FF00AA}{(4)} \end{array} \end{array}
Eliminamos xx
−21x−28y=−721x−18y=99−46y=92\begin{array}{r} -21x - 28y = -7 \\ 21x - 18y = 99 \\ \hline -46y = 92 \end{array} y=92−46y = \dfrac{92}{-46} y=−2y = -2
Eliminamos yy
18x+24y=628x−24y=13246x=138\begin{array}{r} 18x + 24y = 6 \\ 28x - 24y = 132 \\ \hline 46x = 138 \end{array} x=13846x = \dfrac{138}{46} x=3x = 3

Comprobación

3(3)+4(−2)=19−8=11=1\begin{aligned} 3(3) + 4(-2) &= 1 \\ 9 - 8 &= 1 \\ 1 &= 1 \end{aligned}
7(3)−6(−2)=3321+12=3333=33\begin{aligned} 7(3) - 6(-2) &= 33 \\ 21 + 12 &= 33 \\ 33 &= 33 \end{aligned}
Ejercicio 4
(−1)(4){4x−2y=22x−8y=13}(−8)(2)\begin{array}{ccc} \begin{array}{r} \color{#7B2CBF}{(-1)} \\ \color{#7B2CBF}{(4)} \end{array} & \left\{\begin{array}{l} 4x - 2y = 22 \\ x - 8y = 13 \end{array}\right\} & \begin{array}{l} \color{#FF00AA}{(-8)} \\ \color{#FF00AA}{(2)} \end{array} \end{array}
Eliminamos xx
−4x+2y=−224x−32y=52−30y=30\begin{array}{r} -4x + 2y = -22 \\ 4x - 32y = 52 \\ \hline -30y = 30 \end{array} y=30−30y = \dfrac{30}{-30} y=−1y = -1
Eliminamos yy
−32x+16y=−1762x−16y=26−30x=−150\begin{array}{r} -32x + 16y = -176 \\ 2x - 16y = 26 \\ \hline -30x = -150 \end{array} x=−150−30x = \dfrac{-150}{-30} x=5x = 5

Comprobación

4(5)−2(−1)=2220+2=2222=22\begin{aligned} 4(5) - 2(-1) &= 22 \\ 20 + 2 &= 22 \\ 22 &= 22 \end{aligned}
(5)−8(−1)=135+8=1313=13\begin{aligned} (5) - 8(-1) &= 13 \\ 5 + 8 &= 13 \\ 13 &= 13 \end{aligned}

Ejercicios extra

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Bloque 1
{2x+3y=124x−5y=2\begin{cases} 2x + 3y = 12 \\ 4x - 5y = 2 \end{cases}
{5x−2y=163x+4y=−6\begin{cases} 5x - 2y = 16 \\ 3x + 4y = -6 \end{cases}
{3x+5y=172x−3y=−14\begin{cases} 3x + 5y = 17 \\ 2x - 3y = -14 \end{cases}
{6x−4y=202x+5y=13\begin{cases} 6x - 4y = 20 \\ 2x + 5y = 13 \end{cases}
{4x+3y=77x−2y=−24\begin{cases} 4x + 3y = 7 \\ 7x - 2y = -24 \end{cases}
Bloque 2
{2x−7y=175x+3y=22\begin{cases} 2x - 7y = 17 \\ 5x + 3y = 22 \end{cases}
{8x+2y=203x−6y=−33\begin{cases} 8x + 2y = 20 \\ 3x - 6y = -33 \end{cases}
{3x−4y=−16x+5y=−28\begin{cases} 3x - 4y = -1 \\ 6x + 5y = -28 \end{cases}
{7x+2y=64x−3y=20\begin{cases} 7x + 2y = 6 \\ 4x - 3y = 20 \end{cases}
{5x+6y=−22x−4y=−20\begin{cases} 5x + 6y = -2 \\ 2x - 4y = -20 \end{cases}
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