Given the linear equation 2x+3y-8-0, write another linear equation in two variables such that the geometrical representation of the pair so formed is :
(i) intersecting lines
(ii) parallel lines
(iii) coincident lines
Answers
Concept :
The general form for a pair of linear equations in two variables x and y is
a1x + b1y + c1 = 0 ,
a2x + b2y + c2 = 0 ,
Where a1, a2, b1, b2, c1, c2 are all real numbers ,a1²+ b1² ≠ 0 & a2² + b2² ≠ 0.
Condition 1: Intersecting Lines
If a 1 / a 2 ≠ b 1 / b 2 , then the pair of linear equations has a unique solution.
Condition 2: Coincident Lines
If a 1 / a 2 = b 1 / b 2 = c 1 / c 2 ,then the pair of linear equations has infinite solutions.
A pair of linear equations, which has a unique or infinite solutions are said to be a consistent pair of linear equations.
A pair of linear equations, which has infinite many distinct common solutions are said to be a consistent pair or dependent pair of linear equations.
Condition 3: Parallel Lines
If a 1/ a 2 = b 1/ b 2 ≠ c 1 / c 2 , then a pair of linear equations has no solution.
A pair of linear equations which has no solution is said to be an inconsistent pair of linear equations.
Solution:
Given:
2x + 3y - 8 = 0...........................................(i)
i) For intersecting lines, a1 /a2 ≠ b1/b2
∴ Any line intersecting with eq i may be taken as
3x + 2y - 9 = 0 or 3x + 2y -7 = 0
ii) For parallel lines , a1 /a2 = b1/b2 ≠ c1/ c2
∴ Any line parallel with eq i may be taken as
6x + 9y +7 = 0 or 2x + 3y - 12 = 0
iii) For coincident lines, a1 /a2 = b1/b2 = c1/c2
∴ Any line coincident with eq i may be taken as
4x + 6y -16 = 0 or 6x + 9y - 24 = 0
Hope this answer will help you…
Some more questions from this chapter :
On comparing the ratios a1a2,b1b2 and c1c2, and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide :
(i) 5x-4y+8=0
7x+6y-9=0
(ii) 9x+3y+12=0
18x+6y+24=0
(iii) 6x-3y+10=0
2x-y+9=0
https://brainly.in/question/5484292
Solve the following systems of equations:
11x+15y+23=0
7x-2y-20=0
https://brainly.in/question/17115866
a) Intersecting lines
Solution: For intersecting line, the linear equations should meet following condition:
a
2
a
1
=
b
2
b
1
For getting another equation to meet this criterion, multiply the coefficient of x with any number and multiply the coefficient of y with any other number. A possible equation can be as follows:
4x+9y−8=0
(b) Parallel lines
Solution: For parallel lines, the linear equations should meet following condition:
a
2
a
1
=
b
2
b
1
=
c
2
c
1
For getting another equation to meet this criterion, multiply the coefficients of x and y with the same number and multiply the constant term with any other number. A possible equation can be as follows:
4x+6y–24=0
(c) Coincident lines
Solution: For getting coincident lines, the equations should meet following conditiओन्
1
For getting another equation to meet this criterion, multiply the whole equation with any number. A possible equation can be as follows:
4x+6y–16=0