Hard Level Worksheet
Very Short Answer Questions (1 Mark Each)
(1) Find the value of k if the equations kx + 2y = 4 and 3x + 6y = 12 are inconsistent. k =
Perfect! For inconsistency:
(2) Determine whether the pair x - y = 1 and 3x - 3y = 3 is consistent or inconsistent.
Correct! Since 3x - 3y = 3 can be written as x - y = 1, these are the same equation.
(3) Write a condition for a pair of linear equations to have no solution.
Excellent! This is the condition for inconsistent equations (parallel lines).
(4) If a pair of linear equations has a unique solution, what will be the condition between the ratios
Perfect! This ensures the lines intersect at exactly one point.
(5) Find whether the pair 2x - 3y = 5 and 6x - 9y = 15 is consistent or inconsistent.
Correct! Since 6x - 9y = 15 is 3 times the first equation, they represent the same line.
Short Answer Questions (2 Marks Each)
Note: Answer each question with steps and explanation, in 2-3 sentences. Write down the answers on sheet and submit to the school subject teacher.
(1) Solve the pair of equations using substitution method: 2x + 3y = 12, x - y = 1.
x =
Excellent! Always verify by substituting back into both equations.
(2) Solve: 3x - 4y = 5, 2x + y = 3.
x =
Great work with fractions! Always simplify your final answer.
(3) Form a pair of linear equations for: "The sum of the numerator and denominator of a fraction is 9. If 1 is added to both numerator and denominator, the fraction becomes
Let numerator = x, denominator = y
Equation 1:
Equation 2:
Perfect translation from word problem to algebraic equations!
(4) Draw the graphs of the equations x + y = 5 and 2x + 2y = 10. What do you observe?
Both equations represent the
Excellent! This means
(5) If the pair ax + by = c and dx + ey = f has no solution, write the condition for a, b, d, e.
Condition:
Perfect! This represents parallel lines that never intersect.
Long Answer Questions (4 Marks Each)
Note: Answer each question with steps and explanation. Write down the answers on sheet and submit to the school subject teacher.
(1) The difference between two numbers is 26 and one number is three times the other. Find the numbers using algebraic method.
Number 1 =
Excellent systematic approach! The numbers are 39 and 13.
(2) Solve the following pair by the elimination method: 0.5x + 1.5y = 4, 1.5x + 0.5y = 5.
y =
Perfect elimination technique! Always clear decimals first.
(3) Two numbers are such that twice the first added to the second gives 19, and the first added to twice the second gives 20. Find the numbers.
x =
Excellent! The numbers are 6 and 7.
(4) A boat takes 3 hours to go 30 km downstream and 5 hours to return upstream. Find the speed of the boat in still water and the speed of the stream.
Let speed of boat = x km/h, speed of stream = y km/h
x =
Perfect application! Boat speed = 8 km/h, Stream speed = 2 km/h.
(5) The income of a person is partly spent on food and partly saved. He spends ₹500 more on food than on savings. The ratio of money spent on food to savings is 3 : 2. Find the amount spent on food and savings using a system of equations.
Solving: Food = ₹
Excellent problem solving! Food ₹1500, Savings ₹1000.
Part B: Objective Questions (1 Mark Each)
Choose the correct answer and write the option (a/b/c/d)
(1) The pair of equations 2x + 3y = 6, 4x + 6y = 12 represents
(a) Two intersecting lines (b) Parallel lines (c) Coincident lines (d) None of the above
Correct! The second equation is exactly twice the first, so they represent the same line.
(2) The solution of 4x + 3y = 24, 2x - y = 4 is
(a) (2, 4) (b) (3, 4) (c) (2, 3) (d) (4, 2)
Correct!
(3) If the lines represented by a pair of equations are coincident, then the equations have
(a) No solution (b) Infinite solutions (c) Unique solution (d) Exactly two solutions
Correct! Coincident lines are the same line, so every point on the line is a solution.
(4) Which of the following pairs has a unique solution?
(a) x + y = 3, 2x + 2y = 6 (b) x - y = 1, 3x - 3y = 3 (c) 2x + y = 7, 3x + 4y = 10 (d) x + 2y = 4, 2x + 4y = 8
Correct! Only this pair has different slopes (
(5) The equations x + 2y = 5 and 2x + 4y = 10 are
(a) Intersecting lines (b) Parallel lines (c) Coincident lines (d) None
Correct! The second equation is exactly twice the first equation.
(6) The equations 3x + 2y = 7, 6x + 4y = 14 are
(a) Dependent (b) Inconsistent (c) Intersecting (d) Unique
Correct! Dependent equations represent the same line (coincident lines).
(7) For which value of k, will the system kx + 2y = 4, 3x + 6y = 12 be inconsistent?
(a) k = 6 (b) k = 3 (c) k = 1 (d) k = 2
Correct! For inconsistency:
(8) A pair of linear equations has a unique solution if
(a)
Correct! Different slopes ensure the lines intersect at exactly one point.
(9) Graph of a pair of inconsistent equations will be
(a) Two intersecting lines (b) Coincident lines (c) Parallel lines (d) Same lines
Correct! Inconsistent equations represent parallel lines that never meet.
(10) If the pair of linear equations has infinitely many solutions, then the lines are
(a) Parallel (b) Coincident (c) Intersecting at one point (d) None
Correct! Coincident lines are the same line, providing infinitely many solutions.
Linear Equations Challenge
Determine whether these statements about pairs of linear equations are True or False:
Linear Equations Quiz
🎉 You Did It! What You've Learned:
By completing this worksheet, you now have a solid understanding of:
(1) System Classification: Identifying consistent, inconsistent, and dependent systems
(2) Solution Methods: Substitution, elimination, and graphical methods for solving pairs of equations
(3) Geometric Interpretation: Understanding parallel, intersecting, and coincident lines
(4) Condition Analysis: Using ratios
(5) Word Problems: Translating real-world scenarios into algebraic equations
(6) Solution Verification: Checking answers by substitution back into original equations
(7) Special Cases: Handling equations with decimals, fractions, and parameters
(8) Problem Solving: Advanced applications involving speed, money, and number problems
Excellent work mastering advanced concepts of linear equations in two variables!