Graphical method of solution of a pair of linear equations in two variables

For two lines in a plane, which statement is false?

• The two lines will intersect at one point

• The two lines will intersect at two points

• The two lines will not intersect

• The two lines will be coincident

Answer explanation

We know that given two lines in a plane, only one of the following three possibilities can happen: (i) The two lines will intersect at one point. (ii) The two lines will not intersect, i.e., they are parallel. (iii) The two lines will be coincident. Hence, option 2 is false

Fill in the blank, If a₁/a₂ = b₁/b₂ = c₁/c₂ for two straight lines in a plane , then the lines will be ____________

• Parallel

• Intersecting

• Coincident

• None of the above

Answer explanation

If the lines represented by the equation a₁x + b₁y + c₁ =0 and a₂x + b₂y + c₂ =0 are coincident then a₁/a₂ = b₁/b₂ = c₁/c₂

Represent the following pair of lines graphically. How many solutions are possible for the following pair of lines? 4x + 3y = 3 12x + 9y = 20

• One ( unique) solution

• Infinite solutions

• No solution

• Exactly Two solutions

Answer explanation

General form is--- a₁x + b₁y + c₁ =0 a₂x + b₂y + c₂ =0 On comparing , a₁ = 4 a₂ = 12 b₁ = 3 b₂ = 9 c₁ = 3 c₂ = 12 On calculating, a₁/a₂ = 4/12 = 1/3 b₁/b₂ = 3/9 c₁/c₂ = 3/20 Hence, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ so, lines are parallel and do not have any solution.

For what value of k, do the equations 5x – 2 y + 8 = 0 and 15x + ky = – 24 represent coincident lines?

• 6

• 4

• Cannot be coincident

• -6

Answer explanation

General form is--- a₁x + b₁y + c₁ =0 a₂x + b₂y + c₂ =0 On comparing , a₁ = 5 a₂ = 15 b₁ = - 2 b₂ = k c₁ = 8 c₂ = 24 On calculating, a₁/a₂ = 5/15 = 1/3 b₁/b₂ = -2/k c₁/c₂ = 8/24 = 1/3 For coincident lines, a₁/a₂ = b₁/b₂ = c₁/c₂ so,1/3 = -2/k or k = - 6

Which pair of equations is represented by the graph given below ?

• 6x - 3y +10 =0 2x - y +9 =0

• 6x - 3y + 12 = 0 2x - y + 4 = 0

• Both of the above

• None of the above

Answer explanation

The graph represents a pair of parallel lines . . Determine the ratios a₁/a₂ , b₁/b₂ , c₁/c₂ and judge which of the options satisfy the condition for parallel lines, a₁/a₂ = b₁/b₂ ≠ c₁/c₂ For option 1 , 6/2 = -3/-1 ≠ 10/9 i.e. 3 = 3 ≠ 10/9 Hence Option 1 is the correct option. Option 2 represents coincident lines.

Given the linear equation 12x - 4y = 20, which of the following linear equations in two variables is such that the geometrical representation of the pair formed is two intersecting lines. 1. 6x + 4y =10 2. 6x - 2y = 10

• 1

• 2

• 1 and 2 both

• Neither

Answer explanation

Equation is 12x - 4y =20 If the other line has to be intersecting then a₁/a₂ ≠ b₁/b₂ For equation 1, a₁/a₂ = 12/6 = 2 b₁/b₂ = -4/4 = - 1 Therefore, equation 1 and given equation form an intersecting pair of lines. For equation 2, a₁/a₂ = 12/6 = 2 b₁/b₂ = -4/-2 = 2 since a₁/a₂ = b₁/b₂ , equation 2 and given equation do not form an intersecting pair of lines So correct option is option 1.

Represent the following pair of lines graphically. How many solutions are possible for the following pair of lines? 5x ─ y = 7 x ─ y = ─ 1

• One (unique) solution

• Infinite solutions

• No solution

• Exactly two solutions

Answer explanation

5x ─ y = 7 x : 1, 2 y : ─ 2, 3 x ─ y = ─ 1 x : 1, 2 y : 2, 3 On plotting these points we find that the lines intersect each other at one point and have a unique solution. So, the correct option is Option 1.