Linear Equations in Two Variables

This chapter extends your knowledge of linear equations in one variable, such as 2x + 5 = 0, to linear equations in two variables. An equation that can be written in the form ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero, is a linear equation in two variables x and y. You will express statements from daily life as such equations, for example, a notebook costing twice as much as a pen gives x = 2y, and write equations such as x = 5 in standard form as 1x + 0y - 5 = 0. You will learn that a solution of such an equation is a pair of values, one for x and one for y, that satisfies it, and that a linear equation in two variables has infinitely many solutions. You will find solutions by choosing a value of one variable and solving for the other, check whether given pairs are solutions, and represent solutions as points on the Cartesian plane.

Detailed Worksheet: Linear Equations in Two Variables Section A - Definitions (10 marks) 1. What is a linear equation in two variables? Write its general form. (2 marks) 2. What is meant by a solution of a linear equation in two variables? (2 marks) 3. How many solutions does a linear equation in two variables have? Explain briefly. (2 marks) 4. Write x = -3 as a linear equation in two variables in standard form and state the values of the coefficients. (2 marks) 5. Is the equation x^2 + y = 4 a linear equation in two variables? Give a reason. (2 marks) Section B - Calculations and Applications (15 marks) 6. Write each equation in the form ax + by + c = 0 and state the values of a, b and c: (i) 2x + 3y = 9.35 (ii) x - y/5 - 10 = 0 (iii) y = 2x (iv) 3x = -7. (3 marks) 7. Find four different solutions of the equation 2x + y = 7. (3 marks) 8. Check which of the following are solutions of x - 2y = 4: (0, 2), (2, 0), (4, 0), (6, 1), (1, 1). (3 marks) 9. Find the value of k if x = 2, y = 1 is a solution of the equation 2x + 3y = k. Then find two more solutions of the resulting equation. (3 marks) 10. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this. If a pen costs Rs 15, find the cost of the notebook. Write two more pairs of possible costs. (3 marks) Section C - Diagrams (10 marks) 11. Make a table of four solutions of the equation x + y = 6 and plot them on the Cartesian plane. What do you notice about the points? (4 marks) 12. Make a table of three solutions of the equation y = 3x and plot them. Does the point (0, 0) satisfy the equation? (3 marks) 13. Plot three solutions of the equation x - y = 2 on the Cartesian plane and check whether the point (5, 3) lies on the same straight line. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. The taxi fare in a city is Rs 50 for the first kilometre and Rs 16 for each subsequent kilometre. Taking the distance travelled as x km and the total fare as Rs y, write a linear equation. Find the fare for 10 km and the distance travelled for a fare of Rs 338. (5 marks) 15. The temperature in Fahrenheit F and Celsius C are related by F = (9/5)C + 32. Find F when C = 30 and find C when F = 95. Is there a temperature that is numerically the same in both scales? Find it. (5 marks) 16. Yamini and Fatima together contributed Rs 100 towards a relief fund. Write a linear equation for this, find three possible pairs of contributions, and explain why the equation has infinitely many solutions but only some are meaningful in this context. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Check every solution by substituting it back into the equation.