Inicio › Matemáticas 1 › Pendiente y ordenada al origen

Matemáticas 1 · Tema 7

Pendiente y ordenada al origen

Forma y=mx+by = mx + b

Ejemplo

Escribir la recta 2x+3y−12=02x+3y-12=0 en la forma pendiente-ordenada al origen y graficarla.

2x+3y−12=02x + 3y - 12 = 0

3y=−2x+12y=−2x+123y=−23x+4\begin{aligned} 3y &= -2x + 12 \\ y &= \frac{-2x + 12}{3} \\ \mathbf{y} &\mathbf{= -\tfrac{2}{3}x + 4} \end{aligned}

m=−23m = -\dfrac{2}{3}  ·  b=4b = 4

x y (0,4) (6,0) (3,2)

Comprueba que tres puntos de la gráfica cumplen la ecuación 2x+3y−12=02x+3y-12=0.

(0,4)(0,4)
2(0)+3(4)−12=012−12=00=0\begin{aligned} 2(0)+3(4)-12 &= 0 \\ 12-12 &= 0 \\ \mathbf{0} &\mathbf{= 0} \end{aligned}
(6,0)(6,0)
2(6)+3(0)−12=012−12=00=0\begin{aligned} 2(6)+3(0)-12 &= 0 \\ 12-12 &= 0 \\ \mathbf{0} &\mathbf{= 0} \end{aligned}
(3,2)(3,2)
2(3)+3(2)−12=06+6−12=00=0\begin{aligned} 2(3)+3(2)-12 &= 0 \\ 6+6-12 &= 0 \\ \mathbf{0} &\mathbf{= 0} \end{aligned}

Ejercicios

Bloque 1
3x+2y−8=03x+2y-8=0
−4x−3y+9=0-4x-3y+9=0
5x−2y+4=05x-2y+4=0
Bloque 2
2x+3y+6=02x+3y+6=0
−5x+2y+8=0-5x+2y+8=0
3x−4y−12=03x-4y-12=0
Bloque 3
4x+3y−6=04x+3y-6=0
5x−4y+12=05x-4y+12=0
−2x−2y+8=0-2x-2y+8=0

Ejercicios resueltos

▼

3x+2y−8=03x + 2y - 8 = 0

2y=−3x+8y=−32x+4\begin{aligned} 2y &= -3x + 8 \\ \mathbf{y} &\mathbf{= -\tfrac{3}{2}x + 4} \end{aligned}
x y (0,4) (2,1) (1,2.5)

Comprobación de puntos para 3x+2y−8=03x+2y-8=0.

(0,4)(0,4)
3(0)+2(4)−8=03(0)+2(4)-8=0
0=0\mathbf{0=0}
(2,1)(2,1)
3(2)+2(1)−8=03(2)+2(1)-8=0
0=0\mathbf{0=0}
(1,2.5)(1, 2.5)
3(1)+2(2.5)−8=03(1)+2(2.5)-8=0
0=0\mathbf{0=0}

−4x−3y+9=0-4x - 3y + 9 = 0

−3y=4x−9y=−43x+3\begin{aligned} -3y &= 4x - 9 \\ \mathbf{y} &\mathbf{= -\tfrac{4}{3}x + 3} \end{aligned}
x y (0,3) (3,-1) (1.5,1)

Comprobación de puntos para −4x−3y+9=0-4x-3y+9=0.

(0,3)(0,3)
−4(0)−3(3)+9=0-4(0)-3(3)+9=0
0=0\mathbf{0=0}
(3,−1)(3,-1)
−4(3)−3(−1)+9=0-4(3)-3(-1)+9=0
0=0\mathbf{0=0}
(1.5,1)(1.5, 1)
−4(1.5)−3(1)+9=0-4(1.5)-3(1)+9=0
0=0\mathbf{0=0}

5x−2y+4=05x - 2y + 4 = 0

−2y=−5x−4y=52x+2\begin{aligned} -2y &= -5x - 4 \\ \mathbf{y} &\mathbf{= \tfrac{5}{2}x + 2} \end{aligned}
x y (0,2) (2,7) (-2,-3)

Comprobación de puntos para 5x−2y+4=05x-2y+4=0.

(0,2)(0,2)
5(0)−2(2)+4=05(0)-2(2)+4=0
0=0\mathbf{0=0}
(2,7)(2,7)
5(2)−2(7)+4=05(2)-2(7)+4=0
0=0\mathbf{0=0}
(−2,−3)(-2,-3)
5(−2)−2(−3)+4=05(-2)-2(-3)+4=0
0=0\mathbf{0=0}

2x+3y+6=02x + 3y + 6 = 0

3y=−2x−6y=−23x−2\begin{aligned} 3y &= -2x - 6 \\ \mathbf{y} &\mathbf{= -\tfrac{2}{3}x - 2} \end{aligned}
x y (0,-2) (3,-4) (-3,0)

Comprobación de puntos para 2x+3y+6=02x+3y+6=0

(0,−2)(0,-2)
2(0)+3(−2)+6=02(0)+3(-2)+6=0
0=0\mathbf{0=0}
(3,−4)(3,-4)
2(3)+3(−4)+6=02(3)+3(-4)+6=0
0=0\mathbf{0=0}
(−3,0)(-3,0)
2(−3)+3(0)+6=02(-3)+3(0)+6=0
0=0\mathbf{0=0}

−5x+2y+8=0-5x + 2y + 8 = 0

2y=5x−8y=52x−4\begin{aligned} 2y &= 5x - 8 \\ \mathbf{y} &\mathbf{= \tfrac{5}{2}x - 4} \end{aligned}
x y (0,-4) (2,1) (1,-1.5)

Comprobación de puntos para −5x+2y+8=0-5x+2y+8=0

(0,−4)(0,-4)
−5(0)+2(−4)+8=0-5(0)+2(-4)+8=0
0=0\mathbf{0=0}
(2,1)(2,1)
−5(2)+2(1)+8=0-5(2)+2(1)+8=0
0=0\mathbf{0=0}
(1,−1.5)(1,-1.5)
−5(1)+2(−1.5)+8=0-5(1)+2(-1.5)+8=0
0=0\mathbf{0=0}

3x−4y−12=03x - 4y - 12 = 0

−4y=−3x+12y=34x−3\begin{aligned} -4y &= -3x + 12 \\ \mathbf{y} &\mathbf{= \tfrac{3}{4}x - 3} \end{aligned}
x y (0,-3) (4,0) (2,-1.5)

Comprobación de puntos para 3x−4y−12=03x-4y-12=0

(0,−3)(0,-3)
3(0)−4(−3)−12=03(0)-4(-3)-12=0
0=0\mathbf{0=0}
(4,0)(4,0)
3(4)−4(0)−12=03(4)-4(0)-12=0
0=0\mathbf{0=0}
(2,−1.5)(2,-1.5)
3(2)−4(−1.5)−12=03(2)-4(-1.5)-12=0
0=0\mathbf{0=0}

4x+3y−6=04x + 3y - 6 = 0

3y=−4x+6y=−43x+2\begin{aligned} 3y &= -4x + 6 \\ \mathbf{y} &\mathbf{= -\tfrac{4}{3}x + 2} \end{aligned}
x y (0,2) (3,-2) (-3,6)

Comprobación de puntos para 4x+3y−6=04x+3y-6=0.

(0,2)(0,2)
4(0)+3(2)−6=04(0)+3(2)-6=0
0=0\mathbf{0=0}
(3,−2)(3,-2)
4(3)+3(−2)−6=04(3)+3(-2)-6=0
0=0\mathbf{0=0}
(−3,6)(-3,6)
4(−3)+3(6)−6=04(-3)+3(6)-6=0
0=0\mathbf{0=0}

5x−4y+12=05x - 4y + 12 = 0

−4y=−5x−12y=54x+3\begin{aligned} -4y &= -5x - 12 \\ \mathbf{y} &\mathbf{= \tfrac{5}{4}x + 3} \end{aligned}
x y (0,3) (4,8) (-4,-2)

Comprobación de puntos para 5x−4y+12=05x-4y+12=0.

(0,3)(0,3)
5(0)−4(3)+12=05(0)-4(3)+12=0
0=0\mathbf{0=0}
(4,8)(4,8)
5(4)−4(8)+12=05(4)-4(8)+12=0
0=0\mathbf{0=0}
(−4,−2)(-4,-2)
5(−4)−4(−2)+12=05(-4)-4(-2)+12=0
0=0\mathbf{0=0}

−2x−2y+8=0-2x - 2y + 8 = 0

−2y=2x−8y=−x+4\begin{aligned} -2y &= 2x - 8 \\ \mathbf{y} &\mathbf{= -x + 4} \end{aligned}
x y (0,4) (4,0) (2,2)

Comprobación de puntos para −2x−2y+8=0-2x-2y+8=0.

(0,4)(0,4)
−2(0)−2(4)+8=0-2(0)-2(4)+8=0
0=0\mathbf{0=0}
(4,0)(4,0)
−2(4)−2(0)+8=0-2(4)-2(0)+8=0
0=0\mathbf{0=0}
(2,2)(2,2)
−2(2)−2(2)+8=0-2(2)-2(2)+8=0
0=0\mathbf{0=0}

Ejercicios extra 1

▼
Bloque A
6x+5y−15=06x+5y-15=0
7x−3y+9=07x-3y+9=0
−3x−4y+8=0-3x-4y+8=0
Bloque B
3x+2y+4=03x+2y+4=0
3x−2y−6=03x-2y-6=0
−8x+3y+9=0-8x+3y+9=0

Ejercicios extra 2

▼
Bloque 1
−5x+4y−8=0-5x+4y-8=0
2x−5y−10=02x-5y-10=0
−4x+6y+12=0-4x+6y+12=0
−3x+8y+16=0-3x+8y+16=0
4x−7y−21=04x-7y-21=0
5x+3y−9=05x+3y-9=0
5x+3y−6=05x+3y-6=0
Bloque 2
−2x+9y+18=0-2x+9y+18=0
5x−10y−20=05x-10y-20=0
3x−7y+14=03x-7y+14=0
−6x+5y+10=0-6x+5y+10=0
7x+6y−12=07x+6y-12=0
10x−9y−27=010x-9y-27=0
−5x+4y+20=0-5x+4y+20=0
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