Pa.Linear Eq Ex 3.1 Solutions
Question - 1 : - Choose the correctanswer from the given four options:
1. Graphically, thepair of equations
6x – 3y +10 = 0
2x – y +9 = 0
represents two lineswhich are
Answer - 1 : -
(A) Intersecting atexactly one point. (B) Intersecting at exactly two points.
(C) Coincident (D)parallel.
Solution:
(D) Parallel
Explanation:
The given equationsare,
6x-3y+10 = 0
dividing by 3
⇒ 2x-y+ 10/3= 0… (i)
And 2x-y+9=0…(ii)
Table for 2x-y+ 10/3 =0,
Table for 2x-y+9=0
Hence, the pair of equations represents twoparallel lines.
Question - 2 : - Choose the correctanswer from the given four options:
2. The pair ofequations x + 2y + 5 = 0 and –3x – 6y+ 1 = 0 have
Answer - 2 : -
(A) a unique solution(B) exactly two solutions
(C) infinitely manysolutions (D) no solution
Solution:
(D) No solution
Explanation:
The equations are:
x + 2y + 5 = 0
–3x – 6y+ 1 = 0
a1 =1; b1 = 2; c1 = 5
a2 =-3; b2 = -6; c2 = 1
a1/a2 =-1/3
b1/b2 =-2/6 = -1/3
c1/c2 =5/1 = 5
Here,
a1/a2 =b1/b2 ≠ c1/c2
Therefore, the pair ofequation has no solution.
Question - 3 : - Choose the correctanswer from the given four options:
3. If a pair of linearequations is consistent, then the lines will be
Answer - 3 : -
(A) parallel (B)always coincident
(C) intersecting orcoincident (D) always intersecting
Solution:
(C) intersecting orcoincident
Explanation:
Condition for a pairof linear equations to be consistent are:
Intersecting lineshaving unique solution,
a1/a2 ≠b1/b2
Coincident ordependent
a1/a2 =b1/b2 = c1/c2
Question - 4 : - Choose the correctanswer from the given four options:
4. The pair ofequations y = 0 and y = –7 has
Answer - 4 : -
(A) one solution (B)two solutions
(C) infinitely manysolutions (D) no solution
Solution:
(D) no solution
Explanation:
The given pair ofequations are y = 0 and y = – 7.
Graphically, bothlines are parallel and have no solution
Question - 5 : - Choose the correctanswer from the given four options:
5. The pair ofequations x = a and y = b graphicallyrepresents lines which are
Answer - 5 : -
(A) parallel (B)intersecting at (b, a)
(C) coincident (D)intersecting at (a, b)
Solution:
(D) intersecting at (a,b)
Explanation:
Graphically in everycondition,
a, b>>0
a, b< 0
a>0, b< 0
a<0, b>0 but a =b≠ 0.
The pair of equationsx = a and y = b graphically represents lines which are intersecting
at (a, b).
Hence, the cases twolines intersect at (a, b).
Question - 6 : - Do the following pair of linear equations have no solution? Justify youranswer.
Answer - 6 : -
(i) 2x + 4y =3
12y + 6x =6
(ii) x =2y
y = 2x
(iii) 3x + y –3 = 0
2x + 2/3y =2
Solution:
The Condition for nosolution = a1/a2 = b1/b2 ≠c1/c2 (parallel lines)
(i) Yes.
Given pair ofequations are,
2x+4y – 3 = 0 and 6x +12y – 6 = 0
Comparing theequations with ax+ by +c = 0;
We get,
a1 =2, b1 = 4, c1 = – 3;
a2 =6, b2 = 12, c2 = – 6;
a1 /a2 =2/6 = 1/3
b1 /b2 =4/12 = 1/3
c1 /c2 =– 3/ – 6 = ½
Here, a1/a2 =b1/b2 ≠ c1/c2, i.eparallel lines
Hence, the given pairof linear equations has no solution.
(ii) No.
Given pair ofequations,
x = 2y or x – 2y = 0
y = 2x or 2x – y = 0;
Comparing theequations with ax+ by +c = 0;
We get,
a1 =1, b1 = – 2, c1 = 0;
a2 =2, b2 = – 1, c2 = 0;
a1 /a2 =½
b1 /b2 =-2/-1 = 2
Here, a1/a2 ≠ b1/b2.
Hence, the given pairof linear equations has unique solution.
(iii) No.
Given pair ofequations,
3x + y – 3 = 0
2x + 2/3 y = 2
Comparing theequations with ax+ by +c = 0;
We get,
a1 =3, b1 = 1, c1 = – 3;
a2 =2, b2 = 2/3, c2 = – 2;
a1 /a2 =2/6 = 3/2
b1 /b2 =4/12 = 3/2
c1 /c2 =– 3/-2 = 3/2
Here, a1/a2 =b1/b2 = c1/c2, i.e coincidentlines
Question - 7 : - Aftab tells his daughter, “Seven years ago, I was seven timesas old as you were then. Also, three years from now, I shall be three times asold as you will be”. Isn’t this interesting? Represent this situationalgebraically and graphically.
Answer - 7 : -
Solutions:
Let the present age of Aftab be ‘x’.
And, the present age of his daughter be ‘y’.
Question - 8 : - The coach of a cricket team buys 3 bats and 6balls for Rs.3900. Later, she buys another bat and 3 more balls of the samekind for Rs.1300. Represent this situation algebraically and geometrically.
Answer - 8 : -
Solutions:
Letus the cost of a bat be ‘Rs x’
And,the cost of a ball be ‘Rs y’
According to the question,
3x+6y = 3900
And x+3y = 1300
For, 3x+6y = 3900
Or x = (3900-6y)/3
The solution table is
| x | 100 | 300 | 700 |
| y | 600 | 500 | 300 |
For, x+3y = 1300
Or x = 1300-3y
The solution table is
| x | 100 | 400 | 700 |
| y | 400 | 300 | 200 |
The graphical representation is as follows.
Question - 9 : - The cost of 2 kg of apples and 1kg of grapeson a day was found to be Rs.160. After a month, the cost of 4 kg of apples and2 kg of grapes is Rs.300. Represent the situation algebraically andgeometrically.
Answer - 9 : -
Solutions:
Let the cost of 1 kg of apples be ‘Rs. x’
And, cost of 1 kg of grapes be ‘Rs. y’
According to the question,
2x+y = 160
And 4x+2y = 300
For, 2x+y = 160 or y = 160−2x, thesolution table is;
For 4x+2y = 300 or y = (300-4x)/2, thesolution table is;
The graphical representation is as follows;
