Case Study 1: Cost of Pens and Pencils
A shopkeeper sells pens at ₹10 each and pencils at ₹5 each. If a customer buys pens worth ₹x and pencils worth ₹y, then the total cost can be expressed as 10x + 5y. To find the possible purchases for ₹50, the situation can be represented by the linear equation 10x + 5y = 50.
Questions:
- The given equation is of the form:
(a) ax + by = c (b) ax + by + c = 0 (c) y=mx+c (d) none
Answer: (a) - Number of solutions of the equation is:
(a) 1 (b) 2 (c) infinite (d) none
Answer: (c) - If x=2, then y=?
(a) 5 (b) 6 (c) 3 (d) 10
Answer: (a) - If y=4, then x=?
(a) 1 (b) 2 (c) 3 (d) 4
Answer: (c) - Graph of the equation is a:
(a) line (b) parabola (c) circle (d) none
Answer: (a)
Case Study 2: School Bus Fare
A school bus charges ₹20 per student for a local trip and ₹30 per teacher. If x students and y teachers travel, total cost is given by 20x+30y. For a trip costing ₹300, we get the equation 20x+30y=300.
Questions:
- The equation is:
(a) linear in one variable (b) linear in two variables (c) quadratic (d) cubic
Answer: (b) - If x=10, y=?
(a) 0 (b) 2 (c) 3 (d) 5
Answer: (c) - If y=5, then x=?
(a) 0 (b) 5 (c) 10 (d) none
Answer: (a) - The graph of this equation is:
(a) line through origin (b) line not through origin (c) parabola (d) none
Answer: (b) - The intercepts are:
(a) (15,0),(0,10) (b) (0,15),(10,0) (c) (0,10),(15,0) (d) (0,5),(5,0)
Answer: (c)
Case Study 3: Railway Ticket Problem
The cost of 1 first-class ticket is ₹100 and of 1 sleeper ticket is ₹60. If a person buys x first-class and y sleeper tickets for ₹1000, the equation is 100x+60y=1000.
Questions:
- General form of linear equation in two variables is:
(a) ax+by+c=0 (b) ax+by=c (c) y=mx+c (d) none
Answer: (a) - If x=4, then y=?
(a) 10 (b) 5 (c) 20 (d) none
Answer: (b) - If y=10, then x=?
(a) 2 (b) 3 (c) 4 (d) none
Answer: (a) - Equation has:
(a) one solution (b) two solutions (c) infinitely many solutions (d) no solution
Answer: (c) - Graph of solutions is:
(a) straight line (b) parabola (c) ellipse (d) circle
Answer: (a)
Case Study 4: Fruit Seller
A fruit seller sells apples at ₹40/kg and oranges at ₹50/kg. If he sells x kg apples and y kg oranges for a total of ₹500, the equation is 40x+50y=500.
Questions:
- The equation 40x+50y=500 can be simplified to:
(a) 4x+5y=50 (b) 8x+5y=100 (c) x+y=10 (d) none
Answer: (a) - If x=5, then y=?
(a) 2 (b) 5 (c) 6 (d) 10
Answer: (a) - If y=6, then x=?
(a) 2.5 (b) 5 (c) 10 (d) none
Answer: (a) - The line intersects axes at:
(a) (0,10),(12.5,0) (b) (0,12.5),(10,0) (c) (0,5),(10,0) (d) none
Answer: (b) - Graph of this equation passes through:
(a) (5,2) (b) (6,5) (c) (10,10) (d) none
Answer: (a)
Case Study 5: Work and Wages
A worker earns ₹500 per day and a supervisor earns ₹700 per day. If a company hires x workers and y supervisors for a total cost of ₹7000, then 500x+700y=7000.
Questions:
- The given equation is:
(a) quadratic (b) linear in two variables (c) cubic (d) none
Answer: (b) - If x=10, y=?
(a) 0 (b) 2 (c) 5 (d) none
Answer: (b) - If y=5, x=?
(a) 0 (b) 5 (c) 10 (d) none
Answer: (a) - Equation reduces to:
(a) 5x+7y=70 (b) 50x+70y=7000 (c) x+y=10 (d) none
Answer: (a) - Graph intersects axes at:
(a) (14,0),(0,10) (b) (0,14),(10,0) (c) (0,10),(14,0) (d) none
Answer: (c)
Case Study 6: Family Budget
A family spends ₹2000 on food and ₹3000 on clothes every month. If in some month food costs ₹x and clothes cost ₹y with a total of ₹10,000, then x+y=10,000.
Questions:
- This is an equation in:
(a) one variable (b) two variables (c) three variables (d) none
Answer: (b) - If x=4000, y=?
(a) 4000 (b) 5000 (c) 6000 (d) none
Answer: (b) - If y=2000, x=?
(a) 6000 (b) 8000 (c) 9000 (d) none
Answer: (b) - Equation graph:
(a) line parallel to x-axis (b) line parallel to y-axis (c) slant line (d) none
Answer: (c) - General solution is:
(a) infinite pairs (x,y) (b) unique pair (c) none (d) only integers
Answer: (a)
Case Study 7: Age Problem
Sum of ages of father and son is 60 years. Let father’s age be x and son’s age be y. Then equation is x+y=60.
Questions:
- General form is:
(a) ax+by+c=0 (b) ax+by=c (c) y=mx+c (d) none
Answer: (a) - If father is 40, son is:
(a) 10 (b) 15 (c) 20 (d) none
Answer: (c) - If son is 18, father is:
(a) 38 (b) 40 (c) 42 (d) none
Answer: (c) - One solution pair is:
(a) (40,20) (b) (35,25) (c) (50,10) (d) all of these
Answer: (d) - Graph intercepts:
(a) (60,0),(0,60) (b) (0,30),(30,0) (c) (0,60),(60,0) (d) none
Answer: (a)
Case Study 8: Business Profit
Profit on selling a chair is ₹150 and on a table is ₹250. If shopkeeper sells x chairs and y tables with total profit ₹5000, then 150x+250y=5000.
Questions:
- Equation simplified is:
(a) 3x+5y=100 (b) 15x+25y=500 (c) x+y=10 (d) none
Answer: (a) - If x=10, y=?
(a) 10 (b) 15 (c) 20 (d) none
Answer: (a) - If y=20, x=?
(a) 10 (b) 20 (c) 25 (d) none
Answer: (a) - Equation has:
(a) finite solutions (b) infinite solutions (c) no solution (d) none
Answer: (b) - Graph intercepts:
(a) (100,0),(0,20) (b) (0,100),(20,0) (c) (20,0),(0,100) (d) none
Answer: (c)
Case Study 9: Exam Scores
A student scores 2 marks for each correct answer and loses 1 mark for each wrong answer. If x correct and y wrong answers give total 20 marks, then 2x−y=20.
Questions:
- Equation type:
(a) linear in one variable (b) linear in two variables (c) quadratic (d) none
Answer: (b) - If x=15, y=?
(a) 10 (b) 5 (c) 20 (d) none
Answer: (b) - If y=10, x=?
(a) 15 (b) 20 (c) 10 (d) none
Answer: (a) - Number of solutions:
(a) one (b) two (c) infinitely many (d) none
Answer: (c) - Graph shape:
(a) line (b) parabola (c) circle (d) none
Answer: (a)
Case Study 10: Mixture Problem
A milkman mixes milk at ₹50/litre and water at ₹0/litre. If mixture has x litres milk and y litres water costing ₹200, then 50x+0y=200.
Questions:
- Simplified equation is:
(a) 50x=200 (b) x=4 (c) both (a)&(b) (d) none
Answer: (c) - Value of x=?
(a) 2 (b) 3 (c) 4 (d) 5
Answer: (c) - Value of y can be:
(a) any real number (b) any non-negative integer (c) zero only (d) none
Answer: (b) - Equation is parallel to:
(a) x-axis (b) y-axis (c) both (d) none
Answer: (b) - The graph represents:
(a) vertical line x=4 (b) horizontal line y=4 (c) slant line (d) none
Answer: (a)