Worksheet: Equation of a Straight Line: General Form

In this worksheet, we will practice finding and writing the equation of a straight line in general form.

Q1:

Write the equation of the line that passes through the points (2,βˆ’2) and (βˆ’2,10) in the form π‘Žπ‘₯+𝑏𝑦+𝑐=0.

  • A4π‘₯+π‘¦βˆ’4=0
  • B4π‘₯+𝑦+4=0
  • C3π‘₯+𝑦+4=0
  • D3π‘₯βˆ’π‘¦βˆ’4=0
  • E3π‘₯+π‘¦βˆ’4=0

Q2:

A line passes through the points (4,3) and (βˆ’2,βˆ’9).

Find the gradient of the line.

Find the coordinates of the point at which the line intercepts the 𝑦-axis.

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

Hence, write the equation of the line in the form π‘Žπ‘₯+𝑏𝑦+𝑐=0.

  • A2π‘₯+𝑦+5=0
  • B2π‘₯βˆ’π‘¦βˆ’5=0
  • C2π‘₯βˆ’π‘¦βˆ’3=0
  • D2π‘₯βˆ’π‘¦+5=0
  • Eβˆ’2π‘₯βˆ’π‘¦βˆ’3=0

Q3:

Write the equation of the line with slope 32 and 𝑦-intercept (0,3) in the form π‘Žπ‘₯+𝑏𝑦+𝑐=0.

  • A3π‘₯βˆ’2𝑦+3=0
  • B3π‘₯βˆ’2π‘¦βˆ’6=0
  • C3π‘₯βˆ’2𝑦+6=0
  • Dπ‘₯βˆ’2𝑦+6=0
  • E3π‘₯βˆ’π‘¦+6=0

Q4:

What are the π‘₯- and 𝑦- intercepts of the line 3π‘₯+2π‘¦βˆ’12=0?

  • A2, 12
  • B3, 12
  • C2, 3
  • D3, 2

Q5:

A line passes through (βˆ’5,βˆ’3) and cuts out a triangle of area 32 with the two coordinate axes. What is its equation?

  • Aπ‘₯+𝑦+8=0
  • Bπ‘₯+π‘¦βˆ’8=0
  • Cπ‘₯βˆ’π‘¦+8=0
  • Dπ‘₯βˆ’π‘¦βˆ’8=0

Q6:

Find all the lines that pass through (3,2) and whose π‘₯- and 𝑦-intercepts combined distances from the origin is 12.

  • A2π‘₯+𝑦+8=0
  • B2π‘₯+π‘¦βˆ’8=0
  • Cπ‘₯βˆ’2π‘¦βˆ’8=0
  • D2π‘₯βˆ’π‘¦+8=0

Q7:

Given that the coordinates of the points 𝐴 and 𝐡 are (5,7) and (βˆ’4,4), respectively, and the point 𝐢 divides 𝐴𝐡 internally in the ratio 2∢1, determine the equation of the straight line passing through the points 𝐢 and 𝐷(βˆ’2,βˆ’2).

  • Aπ‘₯βˆ’7π‘¦βˆ’12=0
  • B7π‘₯βˆ’π‘¦+12=0
  • C7π‘₯+𝑦+12=0
  • D7π‘₯+π‘¦βˆ’12=0

Q8:

Determine the area of the triangle bounded by the π‘₯-axis, the 𝑦-axis, and the straight line 2π‘₯+7𝑦+28=0.

  • A112 area units
  • B14 area units
  • C56 area units
  • D28 area units

Q9:

Determine the equation of the straight line having a direction vector of u=⟨1,βˆ’2⟩, given that the line intersects the positive part of the 𝑦-axis at a point that is 6Β units away from the origin.

  • A2π‘₯βˆ’π‘¦βˆ’6=0
  • Bπ‘₯+2π‘¦βˆ’6=0
  • C2π‘₯βˆ’π‘¦+6=0
  • D2π‘₯+π‘¦βˆ’6=0

Q10:

A line 𝐿 passes through the points (3,3) and (βˆ’1,0). Work out the equation of the line, giving your answer in the form π‘Žπ‘¦+𝑏π‘₯+𝑐=0.

  • Aπ‘¦βˆ’3π‘₯βˆ’3=0
  • B4π‘¦βˆ’3π‘₯+3=0
  • Cβˆ’4𝑦+3π‘₯βˆ’3=0
  • D4π‘¦βˆ’3π‘₯βˆ’3=0
  • E4π‘¦βˆ’π‘₯βˆ’3=0

Q11:

Let 𝐴 be the point (5,βˆ’1) and 𝐡 be the point (βˆ’1,8). Which of the following points is on ⃖⃗𝐴𝐡?

  • A(βˆ’7,3)
  • B(3,βˆ’7)
  • C(7,7)
  • D(9,βˆ’7)
  • E(βˆ’7,9)

Q12:

Determine the equation of the line which cuts the π‘₯-axis at 4 and the 𝑦-axis at 7.

  • A4𝑦+7π‘₯+28=0
  • B7π‘¦βˆ’4π‘₯βˆ’28=0
  • C7𝑦+4π‘₯βˆ’28=0
  • D4π‘¦βˆ’7π‘₯+28=0

Q13:

Find the equation of the line that passes through the points 𝐴(βˆ’10,2) and 𝐡(0,5), giving your answer in the form π‘Žπ‘¦+𝑏π‘₯+𝑐=0.

  • A10𝑦+3π‘₯βˆ’50=0
  • B3𝑦+10π‘₯βˆ’50=0
  • C3𝑦+10π‘₯βˆ’15=0
  • D3π‘¦βˆ’10π‘₯βˆ’15=0
  • E10π‘¦βˆ’3π‘₯βˆ’50=0

Q14:

Given 𝐴(βˆ’3,βˆ’2), 𝐡(0,5), and 𝐢(2,βˆ’6), find the equation of the straight line that passes through the vertex 𝐴 and bisects 𝐡𝐢.

  • A𝑦+3π‘₯βˆ’7=0
  • B8𝑦+3π‘₯βˆ’7=0
  • Cπ‘¦βˆ’3π‘₯+7=0
  • D8π‘¦βˆ’3π‘₯+7=0

Q15:

A straight line has the equation 3π‘¦βˆ’15π‘₯βˆ’12=0. What is the slope of the line?

Q16:

If the slope of the straight line (3π‘Ž+7)π‘₯+4π‘Žπ‘¦+4=0 equals βˆ’1, find the value of π‘Ž.

Q17:

Find the slope of the line βˆ’2π‘₯+3π‘¦βˆ’2=0 and the 𝑦-intercept of this line.

  • Aβˆ’23, 32
  • Bβˆ’23, 1
  • C23, 23
  • D32, 32

Q18:

Which of the following is a Cartesian equation for a straight line that passes through the point (9,βˆ’10) with direction vector (5,βˆ’9)?

  • Aβˆ’9π‘₯βˆ’5𝑦=0
  • Bβˆ’9π‘₯βˆ’5𝑦+31=0
  • Cβˆ’10π‘₯βˆ’9π‘¦βˆ’31=0
  • Dβˆ’10π‘₯βˆ’9𝑦=0

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