Question 1
Form the pair of linear equations in the following problems, and find their solutions graphically.
(i) In a classroom of 20 students, the number of girls is 6 more than the number of boys. Find the number of boys and girls using the graphical method.
Show Solution
Step 1: Form Equations
Let number of girls = x, boys = y
Total students: x + y = 20 ... (1)
Girls are 6 more than boys: x − y = 6 ... (2)
Step 2: Plot Points
For (1): (10, 10), (14, 6)
For (2): (10, 4), (13, 7)
Step 3: Graph Result
The lines intersect at the point (13, 7).
Answer: Number of Girls = 13, Number of Boys = 7.
(ii) 3 notebooks and 2 pens cost ₹65, while 2 notebooks and 3 pens cost ₹60. Find the cost of one notebook and one pen graphically.
Show Solution
Step 1: Form Equations
Let cost of notebook = x, pen = y
3x + 2y = 65 ... (1)
2x + 3y = 60 ... (2)
Step 2: Solve
Solving these equations (intersecting point on graph):
x = 15, y = 10
Answer: Cost of Notebook = ₹15, Cost of Pen = ₹10.
Question 2
Compare ratios a₁/a₂, b₁/b₂, c₁/c₂ to find if lines are intersecting, parallel or coincident.
(i) 2x − 3y + 6 = 0 ; 4x − 6y + 15 = 0
Show Solution
a₁/a₂ = 2/4 = 1/2
b₁/b₂ = −3/−6 = 1/2
c₁/c₂ = 6/15 = 2/5
Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Answer: Parallel Lines (No Solution).
(ii) 3x + 2y − 5 = 0 ; 6x + 4y − 10 = 0
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a₁/a₂ = 3/6 = 1/2
b₁/b₂ = 2/4 = 1/2
c₁/c₂ = −5/−10 = 1/2
Since a₁/a₂ = b₁/b₂ = c₁/c₂
Answer: Coincident Lines (Infinitely many solutions).
(iii) x − 2y + 5 = 0 ; 3x + 4y − 20 = 0
Show Solution
a₁/a₂ = 1/3
b₁/b₂ = −2/4 = −1/2
Since a₁/a₂ ≠ b₁/b₂
Answer: Intersecting Lines (Unique Solution).
Question 3
Find out whether the following pair of linear equations are consistent or inconsistent.
(i) x + y = 3 ; 2x + 2y = 6
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Ratios: 1/2 = 1/2 = 3/6
Lines are coincident.
Answer: Consistent (Dependent).
(ii) 2x − 3y = 5 ; 4x − 6y = 15
Show Solution
Ratios: 2/4 = 1/2; −3/−6 = 1/2; 5/15 = 1/3
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Answer: Inconsistent (Parallel lines).
(iii) x − y = 8 ; 3x − 3y = 16
Show Solution
Ratios: 1/3; −1/−3 = 1/3; 8/16 = 1/2
a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Answer: Inconsistent (Parallel lines).
Question 4
Which of the pairs are consistent/inconsistent? If consistent, solve graphically.
(i) x + 2y = 4 ; 2x + 4y = 12
Show Solution
Ratios: 1/2 = 2/4 ≠ 4/12
Since 1/2 ≠ 1/3, lines are parallel.
Answer: Inconsistent (No Solution).
(ii) x − y = 2 ; 2x + 3y = 9
Show Solution
Ratios: 1/2 ≠ −1/3
Answer: Consistent.
Graphical Solution:
Line 1 passes through (2,0) and (4,2).
Line 2 passes through (0,3) and (4.5,0).
Intersection Point: (3, 1)
Question 5
Half the perimeter of a rectangular playground is 50 m. The length of the playground is 6 m more than its width. Find the length and width.
Word Problem Solution
Show Solution
Step 1: Form Equations
Let Width = w, Length = l
l = w + 6 ... (1)
Half Perimeter (l + w) = 50 ... (2)
Step 2: Solve
Substitute (1) in (2):
(w + 6) + w = 50
2w = 44 → w = 22 m
l = 22 + 6 → l = 28 m
Answer: Length = 28m, Width = 22m.
Question 6
Given 3x − 4y + 6 = 0, write another equation to form a pair that is:
(i) Intersecting Lines
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Condition: a₁/a₂ ≠ b₁/b₂
Example Answer: 2x − 4y + 6 = 0
(ii) Parallel Lines
Show Solution
Condition: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Multiply coefficients by 2, change constant:
Example Answer: 6x − 8y + 10 = 0
(iii) Coincident Lines
Show Solution
Condition: a₁/a₂ = b₁/b₂ = c₁/c₂
Multiply entire equation by 2:
Example Answer: 6x − 8y + 12 = 0
Question 7
Draw graphs of 2x + y = 6 and 2x − y + 2 = 0. Determine the coordinates of vertices of the triangle formed by these lines and the x-axis.
Graph & Coordinates
Show Solution
Step 1: Find Intercepts
Line 1 (2x + y = 6): Cuts x-axis at (3, 0).
Line 2 (2x − y + 2 = 0): Cuts x-axis at (−1, 0).
Step 2: Find Intersection
Adding equations: 4x = 4 → x = 1.
Put x=1 in Eq 1: 2(1) + y = 6 → y = 4.
Intersection vertex: (1, 4).
Answer: The vertices are (1, 4), (−1, 0), and (3, 0).