Pair of Linear Equations in Two Variables

Many everyday problems involve two unknown quantities linked by two conditions, such as the cost of a pencil and a pen, or the ages of a father and his daughter. A linear equation in two variables has the form ax + by + c = 0, where a and b are not both zero, and its graph is a straight line. Two such equations together form a pair of linear equations, and a solution is a pair of values of x and y that satisfies both. In this chapter you will first solve such pairs graphically, where the two lines may intersect at one point (a unique solution, consistent pair), coincide (infinitely many solutions, dependent pair) or be parallel (no solution, inconsistent pair). You will learn to predict which case occurs by comparing the ratios a1/a2, b1/b2 and c1/c2. You will then solve pairs algebraically by the substitution method and the elimination method, and use them to model and solve word problems on ages, money, fractions, digits and geometry.

Detailed Worksheet: Pair of Linear Equations in Two Variables Section A - Definitions (10 marks) 1. What is a pair of linear equations in two variables? Write its general form. (2 marks) 2. What is meant by a solution of a pair of linear equations? When is a pair called consistent? (2 marks) 3. State the three possible graphical situations for a pair of linear equations and the number of solutions in each case. (2 marks) 4. Write the conditions on the ratios a1/a2, b1/b2 and c1/c2 for a pair of linear equations to have a unique solution, no solution and infinitely many solutions. (2 marks) 5. Briefly describe the substitution method and the elimination method of solving a pair of linear equations. (2 marks) Section B - Calculations and Applications (15 marks) 6. Solve by the substitution method: x + y = 14 and x - y = 4. (3 marks) 7. Solve by the elimination method: 3x + 4y = 10 and 2x - 2y = 2. (3 marks) 8. Find the value of k for which the pair 2x + 3y = 7 and (k - 1)x + (k + 2)y = 3k has infinitely many solutions. (3 marks) 9. 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and that of one pen. (3 marks) 10. A fraction becomes 1 if 1 is added to the numerator and 1 is subtracted from the denominator. It becomes 1/2 if only 1 is added to the denominator. Find the fraction. (3 marks) Section C - Diagrams (10 marks) 11. Draw the graphs of x + 3y = 6 and 2x - 3y = 12 on the same axes. Find the solution from the graph and the coordinates of the vertices of the triangle formed by the two lines and the y-axis. (4 marks) 12. 10 students of Class X took part in a Mathematics quiz. The number of girls is 4 more than the number of boys. Form the pair of equations and solve it graphically. (3 marks) 13. Draw the graphs of 2x + 4y = 10 and 3x + 6y = 12 on the same axes. Describe the lines and state whether the pair is consistent or inconsistent. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. 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, solve it, and verify your answer against both conditions. (5 marks) 15. Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden. Then explain what would happen to the equations, graphically and algebraically, if the problem had said that the length is 4 m more than the width and also 4 m less than the width. (5 marks) 16. A pair of equations is x - 2y = 0 and 3x + 4y = 20. Show without solving that it has a unique solution, then solve it by both substitution and elimination. Which method did you find easier for this pair, and why? (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Draw all graphs on squared paper with a suitable scale and label each line with its equation.