Pair of Linear Equations in Two Variables
Mathematics Class Tenth
NCERT Exercise 3.2
Question: 1. Form the pair of linear equations in the following problems, and find their solutions graphically.
(i) 10 students of class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
Solution:
Let the number of girls `= x`
And the number of boys `=y`
Thus, according to question,
`x+y = 10` and
`x-y = 4`
Now, for `x+y =10`
`=> x = 10 - y`
| `x` | 5 | 4 | 6 |
| `y` | 5 | 6 | 4 |
And for `x-y = 4`
`=> x= 4+y`
| `x` | 5 | 4 | 6 |
| `y` | 1 | 0 | – 1 |
Graphical representation of given pair of equations
Thus, Number of girls = 7 and number of boys = 3. Answer
(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and that of one pen.
Solution:
Let, cost of one pencil = Rs `x`
And cost of one pen = Rs `y`
Therefore, according to question,
`5x + 7 y = 50` ------(i)
And, `7x + 5 y = 46` ------(ii)
For equation (i)
| `x` | 3 | 10 | –4 |
| `y` | 5 | 0 | 10 |
For eqution (ii)
| `x` | 8 | 3 | –2 |
| `y` | –2 | 5 | 12 |
Graphical representation of given pair of equations
Thus, cost of one pencil = Rs. 3 and cost of one pen = Rs. 5 Answer
Question: 2. On comparing the ratios `a_1/a_2, b_1/b_2` and `c_1/c_2`, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident:
(i) `5x-4y +8 =0`
`7x+6y -9 =0`
Solution:
Given,
`5x-4y +8 =0` -------(i)
`7x+6y -9 =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 = 5, b_1 = -4`, and `c_1=8`
And, `a_2 = 7, b_2 = 6` and `c_1 = -9`
Now, `a_1/a_2 = 5/7`
And, `b_1/b_2 = (-4)/7`
Here, since, `a_1/a_2 !=b_1/b_2`, therefore, lines representing given pair of equation has a unique solution and intersects exactly at one point. Answer
(ii) `9x+3y+12=0`
`18x+6y+24 =0`
Solution:
Given, `9x+3y+12=0` ----(i)
`18x+6y+24 =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 = 9, b_1=3` and `c_1 = 12`
And, `a_2 = 18, b_2 = 6` and `c_2 = 24`
Now, `a_1/a_2 = 9/18 = 1/2`
And, `b_1/b_2 = 3/6 = 1/2`
And, `c_1 /c_2 = 12/24 = 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 thus, equations has infinite many solutions. Answer
(iii) `6x-3y+10 = 0`
`2x-y+9=0`
Solution:
Given, `6x-3y+10 = 0` ---------(i)
`2x-y+9=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 = 6, b_1 = -3` and `c_1 = 10`
And, `a_2 = 2, b_2 = -1` and `c_1 =9`
Now, `a_1/a_2 = 6/2 =3`
`b_1/b_2 = (-3)/(-1) = 3`
`c_1/c_2 = 10/9`
Here, since, `a_1/a_2=b_1/b_2 !=c_1/c_1`
Therefore, lines representing given pair of linear equations are parallel, i.e. do not intersect at any point and has no solution. Answer
Question: 3. On comparing the ratios `a_1/a_2, b_1/b_2` and `c_1/c_2`, find out whether the following pair of linear equations are consistent, or inconsistent.
(i) `3x+2y = 5`; `2x-3y=7`
Solution:
Given, `3x+2y = 5`
`=> 3x+2y - 5 =0` --------(i)
And, `2x-3y = 7`
`=>2x-3y - 7=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 = 3, b_1 = 2, c_1 =-5`
And, `a_2 = 2, b_2 = -3, c_2 = -7`
Now, `a_1/a_2 = 3/2`
And, `b_1 /b_2 = 2/(-3)`
And, `c_1/c_2 = (-5)/(-7)`
Here, since `a_1/a_2!=a_2/b_2`
Therefore, given pair of linear equations has exactly one unique solution, and thus given pair of linear equations is consistent. Answer
(ii) `2x-3y =8`; `4x-6y =9`
Solution:
Given, `2x-3y =8`
`=> 2x-3y - 8=0` --------(i)
And `4x-6y =9`
`=> 4x-6y - 9 =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 = -3, c_1 = -8`
And, `a_2 = 4, b_2 = -6, c_2 = -9`
Now, `a_1/a_2 = 2/4 = 1/2`
`b_1/b_2 = (-3)/(-6) = 1/2`
And, `c_1 /c_2 = (-8)/(-9)`
Here, since `a_1/a_2 = b_1/b_2 != c_1/c_2`
Therefore, lines representing given pair linear equations are parallel to each other and thus have no possible solution.
Thus, given pair of linear equations in two variables is inconsistent. Answer
(iii) `3/2x+5/3y=7`;
`9x-10y=14`
Solution:
Given,
`3/2x+5/3y=7`
`=> 3/2x +5/3 y - 7 = 0` -----------(i)
`9x-10y=14`
`=> 9x-10y-14 = 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 = 3/2, b_1 = 5/3, c_1 = -7`
And, `a_2 = 9, b_2 = -10, c_2 = -14`
Now, `a_1/a_2 = (3//2)/9=3/(2xx9)`
`=3/18 = 1/6`
`b_1/b_2 = (5//3)/(-10)`
` = 5/(3xx(-10)) = 5/(-30)=1/(-6)`
`c_1/c_2 = (-7)/(-14) = 1/2`
Here, since, `a_1/a_2 !=b_1/b_2`,
Therefore, given pair of linear equation has a unique solution.
And hence given pair of linear equation is consistent.
(iv) `5x-3y=11`;
` - 10+6y = -22`
Solution:
Given, `5x-3y=11`
`=> 6x-3y - 11 = 0` ---------(i)
And, `- 10+6y = -22`
`=> -10+6y +22 = 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 = 5, b_1 = -3, c_1 = -11`
And, `a_2 = -10, b_2 = 6, c_2 = 22`
Now,
`a_1 /a_2 = 5/(-10) = (-1)/2`
`b_1/b_2 = (-3)/6 = (-1)/2`
`c_1 /c_2 = 11/(-22) = (-1)/2`
Here, since, `a_1/a_2=b_1/b_2=c_1/c_2`,
Therefore, given pair of linear equations has coincident pair of lines and thus have infinite number possible solutions.
Thus, given pair of linear equation is consistent (dependent).
(v) `4/3x+2y = 8`;
`2x+3y = 12`
Solution:
Given,
`4/3x+2y = 8`
`=> 4/3x+2y - 8=0` ----------(i)
And, `2x+3y = 12`
`=> 2x+3y -12 = 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 = 4/3, b_1 = 2, c_1 = -8`
And, `a_2 = 2, b_2 = 3, c_2 = -12`
Now,
`a_1/a_2 = (4//3)/2 = 4/(3xx2) = 2/3`
`b_1/b_2 = 2/3`
`c_1/c_2 = (-8)/(-12) = 2/3`
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. Answer.
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