Pair of Linear Equations in Two Variables
Mathematics Class Tenth
NCERT Exercise 3.2-part-2
Question: 4. Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:
(i) `x+y = 5`,
`2x+2y = 10`
Solution:
Given, `x+y = 5`
`=> x+y ? 5 = 0` ---------(i)
`2x+2y = 10`
`=> 2x+2y ? 10 = 0` ---------(ii)
By comparing given equation (i) with `a_1x+b_1y+c_1=0` and equation (ii) with `a_2x+b_2y+c_2=0`, we get
`a_1 = 1, b_1 = 1, c_1 = -5` And
`a_2 = 2, b_2 = 2, c_2 = -10`
Now,
`a_1/a_2 = 1/2`
`b_1/b_2 = 1/2`
And, `c_1/c_2 = (-5)/(-10) = 1/2`
Here, since, `a_1/a_2 = b_1/b_2 = c_1/c_2`
Therefore, lines representing given pair of linear equations are coincident and has infinitely possible many solutions.
Thus, given pair of linear equations is consistent.
Now, from equation (i)
`x= y ? 5`
Table of solution for this linear equation
| `x` | 4 | 3 | 2 |
| `y` | 1 | 2 | 3 |
And from equation (ii)
`x = (10-2y)/2`
Table of solution for this linear equation
| `x` | 4 | 3 | 2 |
| `y` | 1 | 2 | 3 |
Graphical representation of given pair of linear equations
(ii) `x- y = 8`;
`3x-3y = 16`
Solution:
Given, `x- y = 8`
`=> x ? y ? 8 = 0` ---------(i)
And, `3x-3y = 16`
`=> 3x ? 3y ? 16 = 0` ---------(ii)
By comparing given equation (i) with `a_1x+b_1y+c_1=0` and equation (ii) with `a_2x+b_2y+c_2=0`, we get
`a_1 = 1, b_1 = -1, c_1 = -8`
And, `a_2 = 3, b_2 = -3, c_2 = -16`
Now,
`a_1/a_2 = 1/3`
`b_1 /b_2 = (-1)/(-3) = 1/3`
`c_1/c_2 = (-8)/(-16)= 1/2`
Here, since, `a_1/a_2 = b_1/b_2 !=c_1/c_2`
Therefore, lines representing given pair of linear equations are parallel to each other and do not intersect one another, and hence equations has no solution.
Thus, given pair of linear equations is inconsistent. Answer
(iii) `2x+y -6 = 0`;
`4x-2y-4=0`
Solution:
Given, `2x+y -6 = 0` ------------(i)
`4x-2y-4=0` ----------(ii)
By comparing given equation (i) with `a_1x+b_1y+c_1=0` and equation (ii) with `a_2x+b_2y+c_2=0`, we get
`a_1 = 2, b_1 = 1, c_1 = -6`
And, `a_2 = 4, b_2 = -2, c_2 = -4`
Thus,
`a_1 /a_2 = 2/4 = 1/2`
`b_1 / b_2 = 1/(-2)`
`c_1 / c_2 = (-6)/(-4) = 3/2`
Here, since, `a_1/a_2 !=b_1/b_2`
Therefore, lines representing the given pair of linear equations intersect each other and equations have a unique solution.
Therefore, given pair of linear equations is consistent.
Now, from equation (i)
`2x+y = 6`
`=> x = (6-y)/2`
Thus, table of solution for above equation
| `x` | 0 | 1 | 2 |
| `y` | 6 | 4 | 2 |
And from equation (ii)
`4x-2y = 4`
`=> x = (4+2y)/2`
| `x` | 1 | 2 | 3 |
| `y` | 0 | 2 | 4 |
Graphical representation of the given pair of linear equations.
(iv) `2x-2y-2 = 0`;
`4x-4y-5=0`
Solution:
Given, `2x-2y-2 = 0` -------(i)
`4x-4y-5=0` --------(ii)
By comparing given equation (i) with `a_1x+b_1y+c_1=0` and equation (ii) with `a_2x+b_2y+c_2=0`, we get
`a_1 = 2, b_1 = -2, c_1 = -2`
And, `a_2 = 4, b_2 = -4, c_2 = -5`
Thus,
`a_1 /a_2 = 2/4 = 1/2`
`b_1/b_2 = (-2)/(-4) = 1/2`
And, `c_1 / c_2 = (-2)/(-5) = 2/5`
Here, since, `a_1/a_2=b_1/b_2!=c_1/c_2`
Therefore, lines representing given pair of linear equations are parallel to each other and thus, have no possible solution.
Hence, the given pair of linear equations is inconsistent.
Question: 5. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.
Solution:
Let, the width of the garden `=x`
And length of the garden `= y`
Thus, according to question,
`x+4 = y`
`=>x ? y + 4 = 0` --------(i)
`=> y ?x = 4` ---------(ii)
And, `x+y = 36` --------(iii)
`=>x+y-36=0` ----------(iv)
By comparing given equation (i) with `a_1x+b_1y+c_1=0` and equation (iv) with `a_2x+b_2y+c_2=0`, we get
`a_1 = 1, b_1 = -1, c_1 = 4`
And, `a_2 = 1, b_2 = 1, c_1 = -36`
Here,
`a_1/a_2 = 1/1`
`b_1/b_2 = (-1)/1 = -1`
Here, since, `a_1/a_2 !=b_1/b_2`,
Therefore, pair of equations formed in the given conditions has a unique solution, and lines formed by them intersect each other.
From equation (i) `y-x = 4`
| `x` | 0 | 8 | 12 |
| `y` | 4 | 12 | 16 |
And from equation (ii) `x+y = 36`
| `x` | 0 | 36 | 16 |
| `y` | 36 | 0 | 20 |
Since, lines representing linear equations formed in the given conditions at (16, 20), thus, width of the garden is equal to `16 \ m` and length of the garden is equal to `20\ m`. Answer
Question: 6. 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
Solution:
Given, linear equation is `2x+3y-8=0`
(i) Intersecting lines
Lines representing a pair of linear equation intersect each other only when,
`a_1/a_2 != b_1/b_2`
Here, given one equation, `2x+3y-8=0`
Here, `a_1 = 2, b_1 = 3`
Thus, `a_1/a_2 = 2/2 = 1`
And, `b_1/b_2 = 3/4`
Since, here, `a_1/a_2 != b_1/b_2`
Thus, one possible equation so formed will be
`2x + 4 y ? 8 = 0` Answer
(ii) parallel lines
Lines representing a pair of linear equation are parallel only when,
`a_1/a_2 = b_1/b_2 !=c_1/c_2`
Here, given one equation, `2x+3y-8=0`
Here, `a_1 = 2, b_1 = 3, c_1 = (-8)`
Thus, `a_1/a_2 = 2/2 = 1`
And, `b_1/b_2 = 3/3 = 1`
And, `c_1/c_2 = (-8)/(-12)`
Thus, `a_2 = 2, b_2 = 3` and `c_3 = -12`
Thus, second possible linear equation so formed line of which is parallel to line representing given equation will be
`2x+3y ? 12 = 0` Answer
(iii) coincident lines
Lines representing a pair of linear equations in two variables coincident only when,
`a_1/a_2 = b_1/b_2 = c_1/c_2`
Here, given equation, is `2x+3y-8=0`
Thus, `a_1/a_2 = 2/4 = 1/2`
`b_1/b_2 = 3/6 = 1/2`
And, `c_1 /c_2 = -8/-16 = 1/2`
Thus, second possible equation so formed to satisfy the given condition will be
`4x+6y-16 = 0` Answer
Question: 7. Draw the graphs of the equations `x-y+1 =0` and `3x+2y-12=0`. Determine the coordinates of the vertices of the triangle formed by these lines and the `x-`axis, and shade the triangular region.
Solution:
Given,
`x-y+1 =0` -----------(i)
And, `3x+2y-12 = 0` --------(ii)
From equation (i)
`x = y ? 1`
| `x` | 0 | 1 | 2 |
| `y` | 1 | 2 | 3 |
From equation (ii)
`x = (12-2y)/3`
| `x` | 4 | 2 | 0 |
| `y` | 0 | 3 | 6 |
Thus, graphical representation of the given pair of linear equation:
According to graph, vertices of triangle formed because of given pair of linear equations, are (2, 3), (– 1, 0) and (4, 0). Answer
Reference: