Graphically, solve the pair: \(2x + y = 6\) and \(2x - y + 2 = 0\). Also find the ratio of the areas of the two triangles formed by the pair of lines with (a) the x–axis and (b) the y–axis.
Solution: \(x = 1\), \(y = 4\). Area ratio (with x–axis : with y–axis) = 4 : 1.
Determine, graphically, the vertices of the triangle formed by the lines \(y = x\), \(3y = x\), and \(x + y = 8\).
Vertices: \((0,0)\), \((4,4)\), and \((6,2)\).
Draw the graphs of \(x = 3\), \(x = 5\), and \(2x - y - 4 = 0\). Find the area of the quadrilateral formed by these lines and the x–axis.
Area \(= 8\) square units.
The cost of 4 pens and 4 pencil boxes is Rs 100. Also, three times the cost of a pen is Rs 15 more than the cost of a pencil box. Form the pair of linear equations and find both costs.
Pen = Rs 10 each; Pencil box = Rs 15 each.
Determine, algebraically, the vertices of the triangle formed by the lines \(3x - y = 3\), \(2x - 3y = 2\), and \(x + 2y = 8\).
Vertices: \((1,0)\), \((4,2)\), and \((2,3)\).
Ankita travels 14 km partly by rickshaw and partly by bus. She takes 30 minutes if 2 km is by rickshaw and the rest by bus. If 4 km is by rickshaw and the rest by bus, she takes 9 minutes longer. Find the speeds of the rickshaw and the bus.
Rickshaw speed = 10 km/h; Bus speed = 40 km/h.
A person rows at 5 km/h in still water. It takes thrice as much time to go 40 km upstream as 40 km downstream. Find the speed of the stream.
Speed of stream = 2.5 km/h.
A motor boat covers 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find its speed in still water and the speed of the stream.
Boat in still water = 10 km/h; Stream = 4 km/h.
A two-digit number equals \(8\) times the sum of its digits minus \(5\), and also equals \(16\) times the difference of its digits plus \(3\). Find the number.
The number is 83.
A reserved first-class full ticket from A to B costs Rs 2530. A reserved full + a reserved half together cost Rs 3810. The reservation charge is the same for both, but a half ticket has half fare. Find the full fare and the reservation charge.
Full fare = Rs 2500; Reservation charge = Rs 30 per ticket.
A shopkeeper sells a saree at 8% profit and a sweater at 10% discount to get Rs 1008 in total. If instead she sells the saree at 10% profit and the sweater at 8% discount, she gets Rs 1028. Find the cost price of the saree and the list price of the sweater.
Saree (cost price) = Rs 600; Sweater (list price) = Rs 400.
Susan invests in two schemes A (8% p.a.) and B (9% p.a.). She receives Rs 1860 interest in total. If interchanged, the interest would be Rs 20 more. Find the amounts invested in each scheme.
Scheme A: Rs 12,000; Scheme B: Rs 10,000.
Vijay sold bananas in two lots A and B. For A: Rs 2 for 3 bananas; for B: Re 1 each. Total Rs 400. If he had sold A at Re 1 each and B at Rs 4 for 5 bananas, total would be Rs 460. Find the total number of bananas.
Total bananas = 500 (Lot A: 300, Lot B: 200).