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Given points and determine the pairs of lines are perpendicular: b) 3y = x+3 and y+3x=1, d) y+4x=2 and y−4x=1, f) y=5x+3 and y=-5x+1. 3 on the same Cartesian plane. -2). Another line L2 passes through perpendicular to L2, draw the two lines at the point of intersection.

Mathematics

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3 parts
b

Mistari ni wima\text{Mistari ni wima}

d

Mistari si wima\text{Mistari si wima}

f

Mistari si wima\text{Mistari si wima}

Step-by-step solution

12 steps

Ili kubaini kama jozi za mistari ni wima (perpendicular), tunahitaji kulinganisha miteremko yao. Mistari miwili ni wima ikiwa bidhaa ya miteremko yao ni −1-1.

Part b· 4 steps

3y=x+33y = x + 3 na y+3x=1y + 3x = 1

  1. Pata mteremko wa mstari wa kwanza.

    Badilisha 3y=x+33y = x + 3 kuwa umbo la y=mx+cy = mx + c.

    y=13x+33y = \frac{1}{3}x + \frac{3}{3} y=13x+1y = \frac{1}{3}x + 1

    Mteremko wa mstari wa kwanza ni m1=13m_1 = \frac{1}{3}.

  2. Pata mteremko wa mstari wa pili.

    Badilisha y+3x=1y + 3x = 1 kuwa umbo la y=mx+cy = mx + c.

    y=−3x+1y = -3x + 1

    Mteremko wa mstari wa pili ni m2=−3m_2 = -3.

  3. Zidisha miteremko.

    m1⋅m2=13⋅(−3)=−1m_1 \cdot m_2 = \frac{1}{3} \cdot (-3) = -1
  4. Hitimisho.

    Kwa kuwa bidhaa ya miteremko ni −1-1, mistari hii ni wima.
    Mistari ni wima\boxed{\text{Mistari ni wima}}

Part d

Mistari si wima\text{Mistari si wima}

y+4x=2y + 4x = 2 na y−4x=1y - 4x = 1

  1. Pata mteremko wa mstari wa kwanza.

    Badilisha y+4x=2y + 4x = 2 kuwa umbo la y=mx+cy = mx + c.

    y=−4x+2y = -4x + 2

    Mteremko wa mstari wa kwanza ni m1=−4m_1 = -4.

  2. Pata mteremko wa mstari wa pili.

    Badilisha y−4x=1y - 4x = 1 kuwa umbo la y=mx+cy = mx + c.

    y=4x+1y = 4x + 1

    Mteremko wa mstari wa pili ni m2=4m_2 = 4.

  3. Zidisha miteremko.

    m1⋅m2=(−4)⋅(4)=−16m_1 \cdot m_2 = (-4) \cdot (4) = -16
  4. Hitimisho.

    Kwa kuwa bidhaa ya miteremko si −1-1, mistari hii si wima.
    Mistari si wima\boxed{\text{Mistari si wima}}

Part f

Mistari si wima\text{Mistari si wima}

y=5x+3y = 5x + 3 na y=−5x+1y = -5x + 1

  1. Pata mteremko wa mstari wa kwanza.

    Mstari wa kwanza tayari uko katika umbo la y=mx+cy = mx + c.

    y=5x+3y = 5x + 3

    Mteremko wa mstari wa kwanza ni m1=5m_1 = 5.

  2. Pata mteremko wa mstari wa pili.

    Mstari wa pili tayari uko katika umbo la y=mx+cy = mx + c.

    y=−5x+1y = -5x + 1

    Mteremko wa mstari wa pili ni m2=−5m_2 = -5.

  3. Zidisha miteremko.

    m1⋅m2=(5)⋅(−5)=−25m_1 \cdot m_2 = (5) \cdot (-5) = -25
  4. Hitimisho.

    Kwa kuwa bidhaa ya miteremko si −1-1, mistari hii si wima.
    Mistari si wima\boxed{\text{Mistari si wima}}

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