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Matemáticas 3 · Tema 3

Método gráfico

Ejemplo

{x+2y=44x+2y=−2\begin{cases} x + 2y = 4 \\ 4x + 2y = -2 \end{cases}
Primera recta
x+2y=42y=4−xy=4−x2\begin{aligned} x + 2y &= 4 \\ 2y &= 4 - x \\ y &= \dfrac{4-x}{2} \end{aligned} y=2−12xy = 2 - \dfrac{1}{2}x b=2m=ΔyΔx=−12b = 2 \qquad m = \dfrac{\Delta y}{\Delta x} = -\dfrac{1}{2}
Segunda recta
4x+2y=−22y=−2−4xy=−2−4x2\begin{aligned} 4x + 2y &= -2 \\ 2y &= -2 - 4x \\ y &= \dfrac{-2-4x}{2} \end{aligned} y=−1−42xy = -1 - \dfrac{4}{2}x b=−1m=ΔyΔx=−42b = -1 \qquad m = \dfrac{\Delta y}{\Delta x} = -\dfrac{4}{2}
x=−2x = -2y=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}

Ejercicios

Ejercicio 1
{x+2y=8x−2y=−4\begin{cases} x + 2y = 8 \\ x - 2y = -4 \end{cases}
Ejercicio 2
{−x+3y=122x+3y=3\begin{cases} -x + 3y = 12 \\ 2x + 3y = 3 \end{cases}
Ejercicio 3
{−x+3y=−6x−y=4\begin{cases} -x + 3y = -6 \\ x - y = 4 \end{cases}
Ejercicio 4
{2x+y=33x−2y=8\begin{cases} 2x + y = 3 \\ 3x - 2y = 8 \end{cases}

Respuestas

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Ejercicio 1
{x+2y=8x−2y=−4\begin{cases} x + 2y = 8 \\ x - 2y = -4 \end{cases}
Primera recta
x+2y=82y=8−xy=8−x2\begin{aligned} x + 2y &= 8 \\ 2y &= 8 - x \\ y &= \dfrac{8-x}{2} \end{aligned} y=4−12xy = 4 - \dfrac{1}{2}x b=4m=ΔyΔx=−12b = 4 \qquad m = \dfrac{\Delta y}{\Delta x} = -\dfrac{1}{2}
Segunda recta
x−2y=−4−2y=−4−xy=−4−x−2\begin{aligned} x - 2y &= -4 \\ -2y &= -4 - x \\ y &= \dfrac{-4-x}{-2} \end{aligned} y=2+12xy = 2 + \dfrac{1}{2}x b=2m=ΔyΔx=12b = 2 \qquad m = \dfrac{\Delta y}{\Delta x} = \dfrac{1}{2}
x=2x = 2y=3y = 3

Comprobación

(2)+2(3)=82+6=88=8\begin{aligned} (2) + 2(3) &= 8 \\ 2 + 6 &= 8 \\ 8 &= 8 \end{aligned}
(2)−2(3)=−42−6=−4−4=−4\begin{aligned} (2) - 2(3) &= -4 \\ 2 - 6 &= -4 \\ -4 &= -4 \end{aligned}
Ejercicio 2
{−x+3y=122x+3y=3\begin{cases} -x + 3y = 12 \\ 2x + 3y = 3 \end{cases}
Primera recta
−x+3y=123y=12+xy=12+x3\begin{aligned} -x + 3y &= 12 \\ 3y &= 12 + x \\ y &= \dfrac{12+x}{3} \end{aligned} y=4+13xy = 4 + \dfrac{1}{3}x b=4m=ΔyΔx=13b = 4 \qquad m = \dfrac{\Delta y}{\Delta x} = \dfrac{1}{3}
Segunda recta
2x+3y=33y=3−2xy=3−2x3\begin{aligned} 2x + 3y &= 3 \\ 3y &= 3 - 2x \\ y &= \dfrac{3-2x}{3} \end{aligned} y=1−23xy = 1 - \dfrac{2}{3}x b=1m=ΔyΔx=−23b = 1 \qquad m = \dfrac{\Delta y}{\Delta x} = -\dfrac{2}{3}
x=−3x = -3y=3y = 3

Comprobación

−(−3)+3(3)=123+9=1212=12\begin{aligned} -(-3) + 3(3) &= 12 \\ 3 + 9 &= 12 \\ 12 &= 12 \end{aligned}
2(−3)+3(3)=3−6+9=33=3\begin{aligned} 2(-3) + 3(3) &= 3 \\ -6 + 9 &= 3 \\ 3 &= 3 \end{aligned}
Ejercicio 3
{−x+3y=−6x−y=4\begin{cases} -x + 3y = -6 \\ x - y = 4 \end{cases}
Primera recta
−x+3y=−63y=−6+xy=−6+x3\begin{aligned} -x + 3y &= -6 \\ 3y &= -6 + x \\ y &= \dfrac{-6+x}{3} \end{aligned} y=−2+13xy = -2 + \dfrac{1}{3}x b=−2m=ΔyΔx=13b = -2 \qquad m = \dfrac{\Delta y}{\Delta x} = \dfrac{1}{3}
Segunda recta
x−y=4−y=4−xy=4−x−1\begin{aligned} x - y &= 4 \\ -y &= 4 - x \\ y &= \dfrac{4-x}{-1} \end{aligned} y=−4+11xy = -4 + \dfrac{1}{1}x b=−4m=ΔyΔx=11b = -4 \qquad m = \dfrac{\Delta y}{\Delta x} = \dfrac{1}{1}
x=3x = 3y=−1y = -1

Comprobación

−(3)+3(−1)=−6−3−3=−6−6=−6\begin{aligned} -(3) + 3(-1) &= -6 \\ -3 - 3 &= -6 \\ -6 &= -6 \end{aligned}
(3)−(−1)=43+1=44=4\begin{aligned} (3) - (-1) &= 4 \\ 3 + 1 &= 4 \\ 4 &= 4 \end{aligned}
Ejercicio 4
{2x+y=33x−2y=8\begin{cases} 2x + y = 3 \\ 3x - 2y = 8 \end{cases}
Primera recta
2x+y=3y=3−2xy=3−2x1\begin{aligned} 2x + y &= 3 \\ y &= 3 - 2x \\ y &= \dfrac{3-2x}{1} \end{aligned} y=3−21xy = 3 - \dfrac{2}{1}x b=3m=ΔyΔx=−21b = 3 \qquad m = \dfrac{\Delta y}{\Delta x} = -\dfrac{2}{1}
Segunda recta
3x−2y=8−2y=8−3xy=8−3x−2\begin{aligned} 3x - 2y &= 8 \\ -2y &= 8 - 3x \\ y &= \dfrac{8-3x}{-2} \end{aligned} y=−4+32xy = -4 + \dfrac{3}{2}x b=−4m=ΔyΔx=32b = -4 \qquad m = \dfrac{\Delta y}{\Delta x} = \dfrac{3}{2}
x=2x = 2y=−1y = -1

Comprobación

2(2)+(−1)=34−1=33=3\begin{aligned} 2(2) + (-1) &= 3 \\ 4 - 1 &= 3 \\ 3 &= 3 \end{aligned}
3(2)−2(−1)=86+2=88=8\begin{aligned} 3(2) - 2(-1) &= 8 \\ 6 + 2 &= 8 \\ 8 &= 8 \end{aligned}

Ejercicios extra

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Bloque 1
{x+y=5x−y=1\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}
{2x+y=4x−y=−1\begin{cases} 2x + y = 4 \\ x - y = -1 \end{cases}
{x+3y=6x−y=−2\begin{cases} x + 3y = 6 \\ x - y = -2 \end{cases}
{−x+2y=4x+y=5\begin{cases} -x + 2y = 4 \\ x + y = 5 \end{cases}
{2x−y=3x+y=3\begin{cases} 2x - y = 3 \\ x + y = 3 \end{cases}
Bloque 2
{x+2y=−23x−2y=−6\begin{cases} x + 2y = -2 \\ 3x - 2y = -6 \end{cases}
{−x+y=42x+y=1\begin{cases} -x + y = 4 \\ 2x + y = 1 \end{cases}
{x−3y=32x+3y=−12\begin{cases} x - 3y = 3 \\ 2x + 3y = -12 \end{cases}
{3x+y=6x−2y=2\begin{cases} 3x + y = 6 \\ x - 2y = 2 \end{cases}
{x+y=−12x−y=−5\begin{cases} x + y = -1 \\ 2x - y = -5 \end{cases}
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