Exercise 3.1 3 Questions – Forming Equations
Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." Represent this situation algebraically and graphically.
Let Aftab's present age = x, daughter's present age = y
7 years ago: (xâˆ'7) = 7(yâˆ'7) â‡' xâˆ'7 = 7yâˆ'49 â‡' x âˆ' 7y + 42 = 0
3 years later: (x+3) = 3(y+3) â‡' x+3 = 3y+9 â‡' x âˆ' 3y âˆ' 6 = 0
Graphically, these are two lines intersecting at (42, 12). Aftab is 42, daughter is 12.
The coach of a cricket team buys 3 bats and 6 balls for ₹3900. Later, she buys another bat and 3 more balls of the same kind for ₹1300. Represent this situation algebraically and geometrically.
Let cost of 1 bat = ₹x, cost of 1 ball = ₹y
3x + 6y = 3900 â‡' x + 2y = 1300
1x + 3y = 1300 â‡' x + 3y = 1300
Subtracting: y = 0, x = 1300. The lines intersect at one point â†' unique solution.
The cost of 2 kg of apples and 1 kg of grapes on a day was found to be ₹160. After a month, the cost of 4 kg of apples and 2 kg of grapes is ₹300. Represent the situation algebraically and geometrically.
Let cost of apples = ₹x/kg, grapes = ₹y/kg
2x + y = 160 â‡' y = 160 âˆ' 2x
4x + 2y = 300 â‡' 2x + y = 150 â‡' y = 150 âˆ' 2x
These lines are parallel (same slope, different intercept) â†' no solution.
Exercise 3.2 7 Questions – Substitution & Graphical Method
Solve the following pair of linear equations by the substitution method:
(i) x + y = 14, x âˆ' y = 4
From xâˆ'y=4: x = y+4. Substitute in x+y=14: (y+4)+y=14 â‡' 2y=10 â‡' y=5, x=9
(ii) 3x âˆ' y = 3, 9x âˆ' 3y = 9
Second equation is 3 times the first â†' infinite solutions. General solution: y = 3x âˆ' 3
Check consistency: 2x+3y=8, 4x+6y=7
aâ‚/aâ‚‚=2/4=1/2, bâ‚/bâ‚‚=3/6=1/2, câ‚/câ‚‚=8/7
aâ‚/aâ‚‚ = bâ‚/bâ‚‚ ≠câ‚/câ‚‚ â†' Inconsistent (parallel lines, no solution).
On comparing the ratios aâ‚/aâ‚‚, bâ‚/bâ‚‚ and câ‚/câ‚‚, find out whether the lines are intersecting, parallel, or coincident:
(i) 5xâˆ'4y+8=0, 7x+6yâˆ'9=0 â†' 5/7 ≠âˆ'4/6 â†' Intersecting
(ii) 9x+3y+12=0, 18x+6y+24=0 â†' 9/18=3/6=12/24 â†' Coincident
(iii) 6xâˆ'3y+10=0, 2xâˆ'y+9=0 â†' 6/2=âˆ'3/âˆ'1≠10/9 â†' Parallel
Q4: Which of the following pairs of linear equations are consistent/inconsistent? (Solve each case)
(i) x+y=5, 2x+2y=10 â†' consistent (coincident), infinite solutions. (ii) xâˆ'y=8, 3xâˆ'3y=16 â†' inconsistent (parallel). (iii) 2x+yâˆ'6=0, 4xâˆ'2yâˆ'4=0 â†' consistent, solution: x=2, y=2. (iv) 2xâˆ'2yâˆ'2=0, 4xâˆ'4yâˆ'5=0 â†' inconsistent.
Q7: Draw graphs of xâˆ'y+1=0 and 3x+2yâˆ'12=0. Find the vertices of triangle formed by these lines and x-axis.
Vertices: (âˆ'1,0), (4,0), (2,3). Area = ½×5×3 = 7.5 sq units
Exercise 3.3 3 Questions – Elimination Method
Solve by elimination method: (i) x+y=5, 2xâˆ'3y=4 â†' Multiply first by 3: 3x+3y=15. Add: 5x=19, x=19/5, y=6/5
(ii) 3x+4y=10, 2xâˆ'2y=2 â†' From second: xâˆ'y=1. Multiply by 3: 3xâˆ'3y=3. Subtract from 3x+4y=10: 7y=7, y=1, x=2
Q2: Denominator of a fraction is 4 more than twice the numerator. If 6 is subtracted from both numerator and denominator, denominator becomes 12 times the numerator. Find the fraction.
Let numerator = x. Denominator = 2x+4. Fraction = x/(2x+4). (xâˆ'6)/(2xâˆ'2) = 1/12. Cross multiply: 12xâˆ'72 = 2xâˆ'2 â‡' 10x = 70 â‡' x = 7. Fraction: 7/18.
Q3: Boat goes 30 km upstream and 44 km downstream in 10 hrs. In 13 hrs, it goes 40 km upstream and 55 km downstream. Find speed of boat in still water and speed of stream. â†' Speed of boat = 8 km/h, stream = 3 km/h
Exercise 3.4 2 Questions – More on Elimination
Solve: (i) x/3+y/4=11, 5x/6âˆ'y/3=âˆ'7 â†' Multiply by LCM: x=6, y=36
(ii) 2x+3y=11, 2xâˆ'4y=âˆ'24 â†' Subtract: 7y=35, y=5. Then 2x=11âˆ'15=âˆ'4, x=âˆ'2, y=5
Exercise 3.5 4 Questions – Cross-Multiplication Method
Solve: 8x+5y=9, 3x+2y=4
Using cross-multiplication: x/(5×4âˆ'2×9) = y/(9×3âˆ'4×8) = 1/(8×2âˆ'5×3)
x/(20âˆ'18) = y/(27âˆ'32) = 1/(16âˆ'15) â‡' x/2 = y/(âˆ'5) = 1/1
∴ x = 2, y = âˆ'5
Form the pair of linear equations: (i) 5 pencils and 7 pens cost ₹50, 7 pencils and 5 pens cost ₹46 â†' Pencil=₹3, Pen=₹5
Exercise 3.6 2 Questions – Equations Reducible to Linear Form
Solve: (i) 1/2x + 1/3y = 2, 1/3x + 1/2y = 13/6
Let u = 1/x, v = 1/y. Then: u/2+v/3=2 â†' 3u+2v=12. u/3+v/2=13/6 â†' 2u+3v=13.
Solve: u=2, v=3. So x=1/2, y=1/3
Ritu can row downstream 20 km in 2 hours, upstream 4 km in 2 hours. Find speed in still water and speed of stream. â†' 6 km/h, 4 km/h
Exercise 3.7 (Optional) 8 Questions – Mixed Word Problems
Ages of two friends Ani and Biju differ by 3 years. Ani's father Dharam is twice as old as Ani, and Biju is twice as old as his sister Cathy. Cathy and Dharam's ages differ by 30 years. Find ages of Ani and Biju. â†' Ani=18, Biju=15 OR Ani=24, Biju=27
A train covered a certain distance at uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less. If it were 10 km/h slower, it would have taken 3 hours more. Find distance. â†' 600 km
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