Worksheet: General Equation of a Line through the Intersection Point of Two Lines

In this worksheet, we will practice finding the general and vector forms of the equation of a straight line passing through the point of intersection of two lines.

Q1:

Find the equation of the line perpendicular to 6 𝑥 𝑦 + 8 = 0 and passing through the intersection of the lines 4 𝑥 𝑦 3 = 0 and 3 𝑥 + 8 𝑦 1 = 0 .

  • A 1 9 𝑥 3 9 𝑦 + 8 = 0
  • B 2 𝑥 3 𝑦 + 1 = 0
  • C 7 𝑥 4 2 𝑦 1 = 0
  • D 1 7 3 𝑥 1 5 8 𝑦 + 1 0 1 = 0

Q2:

Determine the equation of the line passing through the point of intersection of the two lines whose equations are 5 𝑥 + 2 𝑦 = 0 and 3 𝑥 + 7 𝑦 + 1 3 = 0 while making an angle of 1 3 5 with the positive 𝑦 -axis.

  • A 2 9 𝑥 + 2 9 𝑦 + 3 9 = 0
  • B 2 9 𝑥 + 𝑦 9 1 = 0
  • C 2 9 𝑥 2 9 𝑦 9 1 = 0
  • D 2 9 𝑥 + 2 9 𝑦 3 9 = 0

Q3:

Find the equation of the straight line that passes through the origin and the point of intersection of the two straight lines 𝑥 = 1 7 4 and 𝑦 = 5 .

  • A 𝑦 = 2 0 1 7 𝑥
  • B 𝑦 = 2 0 1 7 𝑥
  • C 𝑦 = 1 7 2 0 𝑥
  • D 𝑦 = 8 5 4 𝑥

Q4:

Find the equation of the straight line that is parallel to the 𝑦 -axis and passes through the point of intersection of the two straight lines 𝑦 = 3 and 𝑥 = 1 1 1 5 𝑦 .

  • A 𝑥 = 1 1 5
  • B 𝑦 = 1 1 5
  • C 𝑥 = 1 1 5
  • D 𝑥 = 1 1 5 𝑦

Q5:

What is the equation of the line passing through 𝐴 ( 1 , 3 ) and the intersection of the lines 3 𝑥 𝑦 + 5 = 0 and 5 𝑥 + 2 𝑦 + 3 = 0 ?

  • A 1 7 𝑥 2 𝑦 + 2 3 = 0
  • B 2 3 𝑥 + 7 𝑦 + 1 7 = 0
  • C 8 𝑥 + 𝑦 + 8 = 0

Q6:

Find the equation of the straight line that passes through the point of intersection of the two straight lines 𝑥 8 𝑦 = 2 and 6 𝑥 8 𝑦 = 1 and parallel to the 𝑦 -axis.

  • A 𝑥 = 1 7
  • B 𝑥 = 1 7
  • C 𝑥 = 1 3 5 6
  • D 𝑥 = 1 7 𝑦

Q7:

Find the equation of the straight line which passes through the point of intersection of the two lines 1 3 𝑥 5 𝑦 = 1 4 and 2 𝑥 + 1 5 𝑦 = 1 1 and is parallel to the straight line 𝑥 + 8 𝑦 = 1 4 .

  • A 𝑦 1 8 𝑥 = 2 1 5 2 9 6
  • B 𝑦 1 8 𝑥 = 2 1 5 2 9 6
  • C 𝑦 + 1 8 𝑥 = 2 1 5 2 9 6
  • D 𝑦 8 𝑥 + 2 1 5 2 9 6 = 0

Q8:

Find the equation of the straight line which passes through the point of intersection of the two lines 4 𝑥 + 1 5 𝑦 = 1 5 and 4 𝑥 + 3 𝑦 = 1 4 and is parallel to the straight line r = 4 , 0 + 𝑘 5 , 4 .

  • A 𝑦 + 4 5 𝑥 = 2 0 3
  • B 𝑦 5 4 𝑥 = 2 0 3
  • C 𝑦 5 4 𝑥 = 2 0 3
  • D 𝑦 4 5 𝑥 + 2 0 3 = 0

Q9:

Find the equation of the vector that is parallel to the 𝑦 -axis and passes through the point of intersection of the two straight lines r = 𝑘 6 , 4 and 3 𝑥 + 5 𝑦 = 5 .

  • A r = 1 , 0 + 𝑘 1 5 , 1 0
  • B r = 1 5 , 1 0 + 𝑘 1 , 0
  • C r = 0 , 1 + 𝑘 1 5 , 1 0
  • D r = 1 5 , 1 0 + 𝑘 0 , 1

Q10:

Find the vector equation of the straight line that passes through the point of intersection of the two straight lines 8 𝑥 𝑦 = 7 and 5 𝑥 3 𝑦 = 2 and the point ( 1 2 , 8 ) .

  • A r = ( 1 2 , 8 ) + 𝑘 ( 1 3 , 7 )
  • B r = ( 1 2 , 8 ) + 𝑘 ( 7 , 1 3 )
  • C r = ( 1 3 , 7 ) + 𝑘 ( 1 2 , 8 )
  • D r = 1 2 , 8 + 𝑘 1 3 , 7

Q11:

Find the 𝑥 -coordinate of the point at which the straight line 3 𝑥 + 9 𝑦 = 0 cuts the 𝑥 -axis.

  • A 1 9
  • B0
  • C3
  • D9

Q12:

Which of the following equations represents a line through the origin?

  • A 6 𝑥 8 𝑦 = 2
  • B 2 𝑥 𝑦 = 6
  • C 8 𝑥 + 𝑦 = 0
  • D 𝑥 + 7 𝑦 = 3

Q13:

The function 𝑓 ( 𝑥 ) is represented by the line 𝐴 𝐵 and the function 𝑔 ( 𝑥 ) is represented by the line 𝑂 𝐴 where the coordinates of 𝐴 are ( 2 , 5 ) . Find the equations of 𝑓 ( 𝑥 ) and 𝑔 ( 𝑥 ) .

  • A 𝑓 ( 𝑥 ) = 2 , 𝑔 ( 𝑥 ) = 5 2 𝑥
  • B 𝑓 ( 𝑥 ) = 5 , 𝑔 ( 𝑥 ) = 5 𝑥 + 2
  • C 𝑓 ( 𝑥 ) = 5 , 𝑔 ( 𝑥 ) = 2 5 𝑥
  • D 𝑓 ( 𝑥 ) = 2 , 𝑔 ( 𝑥 ) = 2 5 𝑥
  • E 𝑓 ( 𝑥 ) = 5 , 𝑔 ( 𝑥 ) = 5 2 𝑥

Q14:

The lines 𝑥 + 𝑦 + 4 = 0 and 𝑟 = ( 1 , 4 ) + 𝐾 ( 2 , 2 ) intersect orthogonally. Find the coordinates of the point of intersection.

  • A ( 2 , 3 )
  • B ( 8 , 8 )
  • C 9 2 , 1 2
  • D ( 0 , 9 )

Q15:

Find the equation of the straight line that passes through the point of intersection of the two lines 5 𝑥 1 4 𝑦 = 6 and 𝑥 1 0 𝑦 = 9 , given that it is perpendicular to the second line.

  • A 𝑦 + 1 0 𝑥 = 2 3 3 1 2
  • B 𝑦 1 0 𝑥 = 2 3 3 1 2
  • C 𝑦 1 0 𝑥 + 2 3 3 1 2 = 0
  • D 𝑦 1 1 0 𝑥 = 2 3 3 1 2

Q16:

Find the point of intersection of the two straight lines 1 1 𝑥 + 2 0 = 0 and 1 1 𝑦 + 2 = 0 .

  • A 2 0 1 1 , 2 1 1
  • B 2 1 1 , 2 0 1 1
  • C 2 0 1 1 , 2 1 1
  • D ( 2 0 , 2 )

Q17:

Find the equation of the straight line that passes through the point of intersection of the two straight lines 9 𝑥 + 2 𝑦 = 1 and 2 𝑥 𝑦 = 1 2 and intersects the negative direction of the 𝑦 -axis at a point that is 10 length units away from the origin.

  • A 𝑦 + 5 1 2 𝑥 = 1 1 0
  • B 𝑦 1 2 5 𝑥 = 1 1 0
  • C 𝑦 + 1 2 5 𝑥 = 1 0
  • D 𝑦 + 1 2 5 𝑥 = 1 0

Q18:

Determine the point of intersection of the two straight lines represented by the equations 𝑥 + 3 𝑦 2 = 0 and 𝑦 + 1 = 0 .

  • A ( 1 , 1 )
  • B ( 1 , 1 )
  • C ( 2 , 4 )
  • D ( 1 , 1 )

Q19:

Find the equation of the straight line, passing through the point of intersection of the two straight lines 𝑦 9 = 0 and 𝑥 𝑦 = 0 , that intersects the positive directions of the coordinate axes in two points at the same distance from the origin.

  • A 𝑥 + 𝑦 = 0
  • B 𝑥 + 𝑦 + 1 8 = 0
  • C 𝑥 𝑦 1 8 = 0
  • D 𝑥 + 𝑦 1 8 = 0

Q20:

What is the equation of the line passing through 𝐴 ( 3 , 5 ) and the intersection of the lines 4 𝑥 + 2 𝑦 + 1 = 0 and 2 𝑥 + 3 𝑦 2 = 0 ?

  • A 3 0 𝑥 + 1 7 𝑦 + 5 = 0
  • B 1 8 𝑥 + 2 3 𝑦 1 3 = 0
  • C 6 𝑥 + 5 𝑦 1 = 0

Q21:

Find the coordinates of the intersection point between a straight line represented by the equation 3 𝑥 9 𝑦 = 9 and the 𝑦 -axis.

  • A
  • B ( 0 , 1 )
  • C ( 0 , 1 )
  • D ( 0 , 9 )
  • E ( 1 , 0 )

Q22:

At which point do the lines 𝑥 = 7 and 1 6 𝑦 = 1 intersect?

  • A ( 7 , 6 )
  • B ( 7 , 6 )
  • C ( 7 , 6 )
  • D ( 7 , 6 )

Q23:

Find the intersection point of the two straight lines r = 𝐾 3 , 5 and 𝑥 + 3 𝑦 = 0 .

  • A ( 1 , 3 )
  • B ( 3 , 1 )
  • C ( 5 , 3 )
  • D ( 0 , 0 )
  • E ( 3 , 5 )

Q24:

Find the intersection of the lines 𝑥 + 1 3 = 7 and 𝑦 1 6 = 0 .

  • A ( 2 0 , 1 6 )
  • B ( 1 6 , 6 )
  • C ( 6 , 1 6 )
  • D ( 1 6 , 2 0 )
  • E ( 6 , 1 6 )

Q25:

At which point do the lines 2 5 𝑥 = 4 and 1 0 𝑦 = 2 intersect?

  • A ( 5 , 2 0 )
  • B ( 2 0 , 5 )
  • C ( 5 , 1 0 )
  • D ( 1 0 , 5 )
  • E ( 2 0 , 1 0 )

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