Linear Equations in One Variable

A linear equation in one variable, such as 2x + 5 = 11, contains a single variable whose highest power is 1. In this chapter you will solve linear equations in which the variable appears on both sides, as in 3x - 7 = 2x + 5, by adding, subtracting, multiplying or dividing both sides by the same number, or equivalently by transposing terms and changing their signs. You will learn to clear fractions and decimals by multiplying both sides by the LCM of the denominators, and to check each solution by substituting it back into the equation. The chapter places great emphasis on applications. You will translate word problems into equations and solve them: finding consecutive numbers with a given sum, ages of people, the sides of rectangles and triangles, the digits of two-digit numbers, numbers of coins and notes, and fractions. Finally, you will reduce equations such as (2x - 3)/(3x + 2) = -2/3 to linear form by cross-multiplication and solve them.

Detailed Worksheet: Linear Equations in One Variable Section A - Definitions (10 marks) 1. What is a linear equation in one variable? Which of these are linear equations in one variable: 2x + 3 = 9, x^2 = 4, 3y - 1 = y + 5, x + y = 7? (2 marks) 2. What is meant by the solution of an equation? Check whether x = 3 is a solution of 2x + 5 = 11. (2 marks) 3. What is transposition? What happens to the sign of a term when it is transposed from one side of an equation to the other? (2 marks) 4. State the operations that can be performed on both sides of an equation without changing its solution. Why must we never divide both sides by zero? (2 marks) 5. Explain, with an example, how an equation of the form (x + 1)/(2x + 3) = 3/8 is reduced to a linear equation. (2 marks) Section B - Calculations and Applications (15 marks) 6. Solve: (i) 3x - 7 = 2x + 5 (ii) 5t - 3 = 3t - 5 (iii) x/2 - 1/5 = x/3 + 1/4. (3 marks) 7. Solve: (i) 3(t - 3) = 5(2t + 1) (ii) (3y + 4)/(2 - 6y) = -2/5 (iii) 0.25(4f - 3) = 0.05(10f - 9). (3 marks) 8. The sum of three consecutive multiples of 8 is 888. Find the multiples. (3 marks) 9. The present ages of Rahul and Haroon are in the ratio 5 : 7. Four years later, the sum of their ages will be 56 years. What are their present ages? (3 marks) 10. The perimeter of a rectangular sheet of paper is 13 cm and its width is 2.75 cm. Find its length. Verify your answer. (3 marks) Section C - Diagrams (10 marks) 11. Draw a sequence of balance diagrams to solve 3x + 2 = x + 8: first remove x from both pans, then remove 2 from both pans, and finally halve both pans. Write the equation shown at each stage and the solution. (4 marks) 12. Draw a rectangle with length (2x + 3) cm and breadth x cm. If its perimeter is 36 cm, form an equation, solve it and mark the actual length and breadth on your diagram. (3 marks) 13. Solve the equations x + 4 = 1, 2x = 5 and 3x - 1 = 8, and mark their solutions on a number line. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. The sum of the digits of a two-digit number is 9. When the digits are interchanged, the new number is 27 more than the original number. Form an equation, find the number and verify your answer. (5 marks) 15. Bansi has 3 times as many two-rupee coins as he has one-rupee coins. If he has Rs 77 in all, how many coins of each kind does he have? Explain how you would check your answer, and why the number of coins must be a whole number. (5 marks) 16. A student solved 4(x - 2) = 2x + 6 as follows: 4x - 2 = 2x + 6, so 2x = 8 and x = 4. Find the error, solve the equation correctly and verify. Also solve x/3 + 1 = 7/15 and check your answer. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show each step of the solution and verify every answer by substitution.