Linear Equations in Two Variables IMPORTANT TERMS, DEFINITIONS AND RESULTS ANIL TUTORIALS SUMMATIVE ASSESSMENT MULTIPLE CHOICE QUESTIONS

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1 Linear Equations in Two Variables IMPORTANT TERMS, DEFINITIONS AND RESULTS l An equation of the form ax + by + c = 0, where a, b, c are real numbers, is called a linear equation in x and y. For example, 3x + y = 9, 4x y = 1 and 3 x y = are linear equations in 4 l (v) Join these points by a straight line and extend it in both the directions. This line is the graph of the given equation. (i) Equation of a line parallel to the y-axis at a distance a from it is x = a. l l l x and y. A linear equation in two variables can be solved in the same way as a linear equation in one variable. The pair of values of x and y which satisfies the given equation is called solution of the equation in two variables. A linear equation in two variables has infinitely many solutions. In order to draw the graph of a linear equation in two variables we may follow the following method : (i) Express y in terms of x. (ii) Choose at least two convenient values of x and find the corresponding values of y, satisfying the given equation. (iii) Write down these values of x and y in the form of a table. (iv) Plot the ordered pairs (x, y) from the table on a graph paper. SUMMATIVE ASSESSMENT (ii) Equation of a line parallel to the x-axis at a distance b from it is y = b. MULTIPLE CHOICE QUESTIONS [1 Mark] 1. On putting x = 4, y = in the equation 3x y k = 0, the value of k is : (b) (c) 11 (d) 11. The equation of the line whose graph passes through the origin, is : x + 3y = 1 (b) x + 3y = 0 (c) x + 3y = 6 (d) none of these 3. If x = 1, y = 4 is a solution of the equation mx y = 6, then the value of m is : 1 (b) (c) (d) 0 4. The general form of a linear equation in two variables is : ax + by + c = 0, where a, b, c are real numbers and a, b 0 A. Important Questions 1 (b) ax + b = 0, where a, b are real numbers and a 0 (c) ax + bx + c = 0, where a, b, c are real numbers and a, b 0 (d) none of these. If the point 3, 1 3 lies on the graph of the equation 3y = ax, then the value of a is : 1 (b) 1 (c) 3 (d) 3 6. The solution of 4x y = is : x =, y = 3 (b) x = 3, y = 7 (c) x = 4, y = 11 (d) all the above

2 7. At x = 0, the value of y in the equation 9.3x + y = π is : p (b) π (c) π (d) π + 8. The graph of y = m is a straight line parallel to : x-axis (b) y-axis (c) both axes (d) none of these 9. The equation of y-axis is : y = 0 (b) x = 0 (c) y = a (d) x = a 10. The solution of the equation x + y = 3 is : (1, ) (b) (1, 1) (c) (, ) (d) (, 3) 11. Which of the following ordered pairs is the solution of the equation 4x 3y = 10? (1, ) (b) ( 1, ) (c) (1, ) (d) none of these 1. The value of x for y = 0 in the equation y = x + 1 is : (b) 1 1 (c) (d) 13. If the point (, 3) lies on the graph of the equation y = ax 7, then the value of a is : 1 (b) 1 (c) 1 1 (d) Any point on the line y = x is of the form : (a, a) (b) (0, a) (c) (a, 0) (d) (a, a) 1. If we multiply or divide both sides of a linear equation with a non-zero number, then the solution of the linear equation : changes (b) remains the same (c) changes in case of multiplication only (d) changes in case of division only 16. The point of the form (a, a) always lies on the line : x = a (b) y = a (c) y = x (d) x + y = The linear equation x y = 7 has : a unique solution (b) two solutions (c) infinitely many solutions (d) no solution 18. The equation x + y = 7 has a unique solution, if x, y are : natural numbers (b) positive real numbers (c) real numbers (d) rational numbers 19. Any point on the y-axis is of the form : (x, 0) (b) (x, y) (c) (0, y) (d) (y, y) 0. The graph of which of the following equations does not pass through the origin? y = x 3 (b) y = mx (c) y = x (d) y = 1 1. If the points ( 1, a), (b, 1) and (c, 0) lie on the straight line with equation y = x, then the values of a, b and c respectively are : 4,, 3 (b), 3, 4 (c) 3, 4, (d) 1, 4,. Which of the following lines is not parallel to y-axis? x = 1 (b) x = 0 (c) x + 3 = 0 (d) y + 1 = 0 3. The linear equation F= C is used to convert the temperature from Fahrenheit to Celsius or vice-versa. The temperature numerically same in both Fahrenheit and Celsius is : 3 (b) 40 (c) 40 (d) none of these 4. If the points (1, 0) and (, 1) lie on the graph of x y +, then the values of a and b are : a b a = 1 and b = 1 (b) a = 1 and b = 1 (c) a = 1 and b = 1 (d) a = 0 and b = 1. The coordinates of the point where the line 3x = y + 4 meets the x-axis are : (3, 0) (b) (0, 3) (c) ( 3, 0) (d) (0, 3) 6. If the points (0, 1) and (1, 0) lie on the graph of the equation y = mx + c, then the values of m and c are : m = 1, c = 1 (b) m = 1, c = 1 (c) m = 1, c = 1 (d) m = 0, c = 1 7. The graph of which of the following is parallel to x-axis? x = a (b) y = a (c) x = 3 + y (d) y = 0 8. The equation whose graph passes through the origin is : x + y = (b) x y = (c) x = 1 y (d) none of these 9. The point where the graphs of the equations x = 3 and y = 3 intersect each other is : (0, 3) (b) (3, 0) (c) (3, 3) (d) none of these 30. The value of y at x = 1 in the equation y = is : (b) (c) 10 (d) Which of the following is not a linear equation in two variables? p + 4q 7 = 0 (b) πu v 3 = 0 ( ) (d) 1.s + 3t 4 = 0 (c) x 7y 3. The force (y) applied on a body is directly proportional to the acceleration (x) produced in the body. The linear equation to represent the given information is :

3 xy = k (b) y = kx (c) x = ky (d) none of these 33. Which of the following is not true about the equation x + 1 = x 3? the graph of the line is parallel to y-axis. (b) x = 4 is the solution of the equation. (c) point ( 4, 0) lies on the line. (d) the graph is at right side of the y-axis. 34. The taxi fare in a city is as follows : for the first kilometre the fare is Rs 10 and for the subsequent distance it is Rs 7 per km. Taking the distance as x km and total fare as Rs y, a linear equation which represents the above information will be : y = x + 10 (b) y = 7x + 3 (c) y = 7x (d) none of these 3. How many linear equations in x and y can be satisfied by x = 1 and y =? only one (b) two (c) infinitely many (d) three 36. The point of the form (a, a) always lies on : x-axis (b) y-axis (c) the line y = x (d) the line x + y = The graph of the linear equation x + 3y = 6 cuts the y-axis at the point : (, 0) (b) (0, 3) (c) (3, 0) (d) (0, ) 1. For the equation x y = 4, check which of the following is a solution. (0, ) (b) (, 0) (c) (4, 0) (d) (1, 1). The equation which represents a line which is parallel to x-axis at a distance of 3 cm from the origin is : x = 3 and x = 3 (b) y = 3 and y = 3 (c) x = 3 only (d) y = 3 only 3. Which of the following is a solution of the equation x + y = 14? x = ; y = 1 (b) x = 0; y = 7 (c) x = ; y = (d) x = 1; y = 3 4. The equation x = 4 represents the line which : is parallel to x-axis (b) passes through origin (c) is parallel to y-axis (c) is perpendicular to y-axis. Equation of the line y = 0 represents : y-axis (b) x-axis (c) both x-axis and y-axis (d) origin 38. The equation x = 7, in two variables, can be written as : 1.x + 1.y = 7 (b) 1.x + 0.y = 7 (c) 0.x + 1.y = 7 (d) 0.x + 0.y = The graph of y = 6 is a line : parallel to x-axis at a distance 6 units from the origin. (b) parallel to y-axis at a distnace 6 units from the origin. (c) making an intercept 6 on the x-axis. (d) making an intercept 6 on both the axes. 40. The positive solutions of the equation ax + by + c = 0 always lie in the : Ist quadrant (b) nd quadrant (c) 3rd quadrant (d) 4th quadrant 41. The graph of the linear equation x + 3y = 6 is a line which meets the x-axis at the point : (0, ) (b) (, 0) (c) (3, 0) (d) (0, 3) 4. The graph of the linear equation y = x passes through the point : 3 3, (b) 0, 3 (c) (1, 1) B. Questions From CBSE Examination Papers 6. Which of the following lines passes through (1, )? x + y = 3 (b) x y = 1 (c) x = (d) x = 1 (d) 1 1, 7. Which of the following is not a solution of the equation x + y = 7? (1, ) (b) (3, 1) (c) (1, 3) (d) (0, 7) 8. Age of x exceeds age of y by 7 years. This statement can be expressed as linear equation as : x + y + 7 = 0 (b) x y + 7 = 0 (c) x y 7 = 0 (d) x + y 7 = 0 9. If (, 0) is a solution of linear equation x + 3y = k, then the value of k is : 4 (b) 6 (c) (d) 10. x =, y = is a solution of the linear equation : x + y = 7 (c) x + y = 7 (b) x + y = 7 (d) x + y = If point (3, 0) lies on the graph of the equation x + 3y = k, then the value of k is : 6 (b) 3 (c) (d) 1. The equation of x-axis is : x + y = 0 (b) x y = 0 (c) y = 0 (d) x = ( 3, ) is a point, which belongs to the graph of the equation : y = x + 1 (b) x = 3y + 1 (c) 3x = y (d) x = y 3

4 14. The graph of equation of the form ax + by + c = 0 where a, b are non-zero numbers, represents : a triangle (b) a ray (c) a straight line (d) a line segment 1. The condition that the equation ax + by + c = 0 represent a linear equation in two variables is : a 0, b = 0 (b) b 0, a = 0 (c) a = 0, b = 0 (d) a 0, b The linear equation y 3 = 0, represented as ax + by + c = 0, has : a unique soluiton (b) infinitely many soluitons (c) two solutions (d) no solution 17. The number of line(s) passing through a point (3, 4) is / are : only one (b) two (c) infinite (d) three 18. The graph of the equation ax + by + c = 0 may be of the form : (c) (b) (d) 19. The graph of the equation x + a = 0 is a line parallel to y-axis and to the left of the y-axis if : a < 0 (b) a = 0 (c) a > 0 (d) for any real value of a 0. The value of k for which x = 1, y = 1 is a solution of kx y = 0 is : 1 (b) (c) (d) 8 1. Equation y = x + 3 has : unique solution (b) no solution (c) only two solutions (d) infinitely many solutions. Equation of line parallel to x-axis and units above the origin is : x = (b) x = (c) y = (d) y = 3. Any point on the line x + y = 0 is of the form : ( a, a) (b) (a, a) (c) (0, a) (d) (a, 0) 4. The linear equation x = in two variables can be written as : 1.x + = 0 (b) 0.x + 1.y + ( ) = 0 (c) 1.x + 0.y + ( ) = 0 (d) 1.x + 1.y + ( ) = 0. A linear equation in two variables has how many solutions? one (b) two (c) infinite (d) not possible 6. The graph of x = 1 is a straight line : intersecting both the axes (b) parallel to y - axis (c) parallel to x-axis (d) passing through the origin 7. The point of the form (a, a) always lies on : x-axis (b) y-axis (c) the line y = x (d) the line x + y = 0 8. Graph of linear equation 4x = in a plane is : parallel to x axis (b) parallel to y-axis (c) lies along x -axis (d) passes through origin 9. The graph of y = mx is a straight line : parallel to x-axis (b) parallel to y-axis (c) passing through origin (d) coincides with x - axis 30. The equation y = x + has : a unique solution (b) no solution (c) infinite solutions (d) only four solutions 31. For what value of k, x = and y = 1 is a solution of x + 3y k = 0? 1 (b) (c) (d) 3 3. The point lying on the equation x y = is : (, 1) (b) (6, 1) (c) ( 3, 1) (d) (3, 4) 33. Point P(, 3) lies on the line represented by the equation : x + y = 0 (b) x + y = 0 (c) x + y = 1 (d) x + y = The graph of the equation of the form y = mx is a line which always passes through : (0, m) (b)(x, 0) (c) (0, y) (d) (0, 0) 3. If a linear equation has solutions (, ), (0, 0) and (, ), then it is of the form : y x = 0 (b) x + y = 0 (c) x + y = 0 (d) x + y = If the line represented by the equation 3x + αy = 8 passes through the points (, ), then the value of α is : 4 (b)1 (c)3 (d)0 4

5 37. The linear equation x + y = 8 has : two solutions (b) a unique solution (c) no solution (d) infinitely many solutions 38. Which of the following represents a line parallel to x-axis? x + y = 7 (b) x + 3 = 0 (c) y + = 3y (d)x + 3 = Any solution of the linear equation x + 0y = 9 in two variables, is of the form : 9, 0 (b) (c) n, 9, n is a real number (d) 0, 9 9, n, n is a real number SHORT ANSWER TYPE QUESTIONS 40. Any point on the x-axis is of the form : (0, y) (b)(x, 0) (c)(x, x) (d)(x, y) 41. Linear equation in one variable is : x = y (b) y = 3y + (c) 4x y = (d)3t + = 9t 7 4. Which of the following is a solution of the equation 4x + 3y = 16? (, 3) (b) (1, 4) (c) (, 4) (d) (1, 3) 43. Graph of the equation x + 3y = 9 cuts y-axis at the point : 9, 0 (b) (0, 9) (c) (0, 3) (d) (3, 1) 44. Which of the following is not a linear equation? ax + by + c = 0 (b) 0x + 0y + c = 0 (c) 0x + by + c = 0 (d) ax + 0y + c = 0 [ Marks] 1. Write whether the following statement is true or false : The coordiantes of points given in the table represent some of the solutions of the equation x + = y x y Express the following equations in the from ax + by + c = 0 and indicate the values of a, b and c. (i) x = y (ii) y = 4x 3. Every point on the graph of a linear equation in two variables does not represent a solution of the linear equation. Is it true? Justify your answer. 4. Check whether the graph represents the linear equation x + y = 0 or not. A. Important Questions 6. Find the value of p from the equation 3x + 4y = p, if its one solution is x =, y = Frame a linear equation in the form ax + by + c = 0 by using the given values of a, b and c. (i) a =, b = 3, c = 4 (ii) a =, b = 0, c = 7 8. Find whether the given ordered pair is a solution of the given linear equation : (i) x 4y = 3; (8, 4) (ii) 4x y = 10; (3, 1) 9. Draw the graph of : (i) x = 4 (ii) y = Check whether the graph represents the linear equation x = 3.. Check whether the point (0, 3) lies on the graph of the linear equation 3x + 4y = Express each of the following equations in the form y = mx + c (i) 3x y = 4 (ii) y x = 1. Draw the graph of 3x y = Check whether the graph of the linear equation x + y = 7 passes through the point (0, 7). 14. Draw the graph of the equation y = x. From the graph, find the value of y when x =.

6 B. Questions From CBSE Examination Papers 1. Draw the graph of the linear equation in two variables : x + y = 3. If the point (, 1) lies on the line x y = k, find k. Also, find one more solution for the given equation. 3. The point (3, 4) lies on the graph of the equation 3y = ax + 7; Find the value of a 4. For what value of k is x =, y = 3 a solution of (k + 1) x (k + 3) y 1 = 0?. Find the co-ordinates of points where the graph of equation 4x + 3y = 1 intersects x-axis and y-axis. 6. The auto fares in a city are as follows. For the first kilometre the fare is Rs 1 and for the subsequent distance it is Rs 7 per km. Taking the distance covered as x km and the total fare as Rs y, write a linear equation. 7. If the point (k 3, k + ) lies on the graph of the equation x + 3y + 1 = 0, find the value of k. 8. Find a value of p for which x =, y = 1 is a solution of the linear equation x + py = p 9. Check which of the following is (are) solution (s) of the equation 3y x = 1 (i) (4, 3) (ii) (, 3 ) 10. Find the value of k so that x = 1 and y = 1 is a solution of the linear equation ikx + 1ky = Give the equation of one line passing through (, 14). How many more such lines are there and why? 1. If x = and y = 1 is the solution of the linear equation x + 3y + k = 0, find the value of k. 13. Express the equation x = y in the general form and indicate the values of a, b and c. 14. How many solution (s) of equation x + 1 = x 3 are there : on number line? (b) in Cartesian plane? 1. Give geometric representation of equation 3x + 1 = 0 in (i) one variable (ii) two variables 16. Find the point at which the equation 3x y = 6 meets the x-axes. 17. Find the coordinates of the points where the line x y = 3 meets both the axes. 18. Find four solutions of x y = Give two solutions of the equation x + 3y = After years, the age of father will be two times the age of son. Write a linear equation in two variables to represent this statement. 1. Express y in terms of x from the equation 3x + y = 8 and check whether the point (4, ) lies on the line.. Express 3x = y in the form of ax + by + c = 0 and hence indicate the values of a, b and c. 3. If the point ( 1, ) lies on the graph of 3x = ay + 7, then find the value of a. 4. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.. Express y in term of x, given that x y = 7. Check whether the point ( 3, ) lies on the given line. 6. Express y in terms of x in equation x 3y = 1. Find the points where the line represented by this equation cuts x-axis and y-axis. SHORT ANSWER TYPE QUESTIONS 1. Find the points where the graph of the equation 3x + 4y = 1 cuts the x-axis and the y-axis.. Draw the graph of the linear equations y = x and y = x on the same cartesian plane. What do you observe? 3. At what point does the graph of the linear equation x + y = meet a line which is parallel to the y-axis, at a distnace of units from the origin in the positive direction of x-axis? 4. Let y varies directly as x. If y = 1 when A. Important Questions 6 x = 4 then write a linear equation. What is the value of y when x =?. Draw a square whose sides are repreented by x = 4, x = 4, y = 4, y = Draw a triangle whose sides are represented by x = 0, y = 0 and x + y = Determine the point on the graph of the linear equation x + y = 19 whose ordinate is 1 1 times of its abscissa. [3 Marks]

7 8. Determine the point on the graph of the linear equation x + y = 0 whose abscissa is 1 times its ordinate. 9. Draw the graph of the equation represented by a straight line which is parallel to the x-axis and at a distnace of 3 units below it. 10. Find three solutions of x y + 6 = 0 after reducing it to y = mx + c form. 11. Draw the graph of the equation represented by the striaght line which is parallel to the x-axis and is 4 units above it. 1. Draw the graph of the linear equation whose solutions are represented by the points having the sum of the coordinates as 10 units. 13. Write the linear equation such that each point on its graph has an ordinate three times its abscissa. 14. For what value of p, the linear equation x + py = 8 has equal values of x and y for its solution? 1. Find the solution of the linear equation x + y = 0 which represents a point on (i) x-axis (ii) y-axis. B. Questions From CBSE Examination Papers 1. Draw the graph of the eqation x 3y =. From the graph, find the value of y when x = 4. Draw the graph of the equation 3x + y = 6. Find the area of the triangle formed with the line, x- axis and y-axis. 3. Find any three solutions for the equation 1x y = Draw the graph x + y = 6 and find the points where the line cuts x-axis and y-axis.. Draw the graph x + y = 4 and find the area of the triangle formed by the line with x-axis and y-axis. 6. The cost of a table exceeds the cost of the chair by Rs 10. Write a linear equation in two variables to represent this statement. Also, find two solutions for the same equation. 7. Draw the graph of the following linear equations : x y = 7 and x = y + 3. At what points does the graph of each equation cut the x-axis? 8. If the point (4, 3) lies on the graph of the equation 3x ay = 6, find whether (, 6) also lies on the same graph. Find the coordinates of the points where the graph cuts x-axis and y-axis. 9. Solve for x : x 1 = 3 4 (x + 1) Draw the graph of equation x y =1 and x + y = 8 on the same axes. Shade the area bounded by these lines and x-axis. 11. Draw the graph of the equation y x = 7 and determine from the graph if x = 3, y = is its solution or not. 1. Observe the graph and answer the following questions : (i) Write the coordinates of point B and C. (ii) Find one more solution of line passing through A and B. (iii) Write equations of x-axis and y-axis. 13. Draw the graph of the equation y = x + 1 and find the point where the graph meets the axes. 14. The linear equation that converts Fahrenheit (F) to Celsius (C) is given by the relation C = ( F 3 ). 9 What is the numerical value of the temperature which is same in both the scales? 1. Draw the graphs of x y = 3 and 3x + y = 1 on the same graph paper. Find the point of intersection of these graphs. 16. If the number of hours for which a labourer works is x and y are his wages (in rupees) and y = x 1, draw the graph of the work-wages equation. From the graph, find the wages of the labourer if he works for 6 hours. 17. Yamini and Fatima, two students, together contribute Rs 100 towards the PM Relief Fund to help the earthquake victims. Write a linear equation which satisfies this data. Draw the graph of the same. 7

8 18. Express the lienar equation = 3x in the form ax + by + c = 0 and indicate the values of a, b and c. Also give the geometrical representation of above equation in two variables. 19. Ashish and Deepak contribute to charity. The contribution of Ashish is of contribution of Deepak. Write a linear equation to represent the above and draw the graph. From the graph, find the contribution of Ashish, if Deepak contributes Rs Solve the equation y + 3 = 3y and represent the solution(s) on : (i) the number line (ii) the cartesian plane 1. In countries like USA and Canada, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Fahrenheit to Celsius. F = 9 C + 3 (i) If the temperature is 30 C, what is the temperature in Fahrenheit? (ii) If the temperature is 9 F, what is the temperature in Celsius? (iii) Find the temperature which is numerically the same in both Fahrenheit and Celsius?. Give geometric representation of y + 7 = 0 as an equation : (i) in one variable (ii) in two variables 3. Draw a graph of linear equation 3x + y = Find two different solutions for the linear equation 3x + y = 1 and check whether (, 3) is the solution.. Find four solutions of the following linear equation in two variables. (x + 3) 3 (y 1) = 0 6. Draw the graph of the equation 3x + 4y = 1 and find the coordinates of the points of intersection of the graph with axes. 7. The taxi fare in a city is charged as per the rates stated below : Rate for the first kilometre of joureney is Rs and the rate for the subsequent distance covered is Rs 4 per km. Taking distance covered as x km and total fare as Rs y, write the linear equation in variables x and y to express the above statement. Draw the graph for the linear equation. 8. Alka and Noori, two students of class IX, together contributed Rs 00 towards Prime Minister s Relief Fund to help earthquake victims. Write a linear equation which satisfies this data and draw the graph of the same. 9. Find the value of a for which the equation x + ay = has (1, 1) as a solution. Find two more soluiotns for the equation obtained. 30. Solve the linear equation for x : x 3 x + 3 x = Find a, if a + 3 a a = +, a, a 7. a Two years later a father will be eight years more than three times the age of the son. Taking the present age of father and son as x and y respectively, Write a linear equation for the above and draw its graph. (b) From the graph, find the age of father when son s age is 10 years. 33. Draw the graph of the equation 3x + y = and write the co-ordinates of the points where the line intersects x-axis and y-axis. 34. Draw the graph of the equaiton x + 3y = 6. Write the points where the line intersects the x-axis and the y -axis. 3. Two pens and three pencils together cost Rs 0. Represent this statement as a linear equation in two variables and give two solutions for it. Also, verify if (4, ) is a solution of the equation formed. 36. Draw the graph of the equation represented by a straight line which is parallel to x-axis and at a distance 3 units below it. Write its equation also. 37. Solve the equation x + 1 = x 3 and represent the solution on : (i) the number line (ii) the cartesian plane 38. Draw the graph of the equation (x + 3) 3 (1 + y) = 0. Also, find the point where the line meets x-axis. 39. Give the geometric representations of x + = 0 as an equation (i) in one variable (ii) in two variables. 40. Express the equation y = x + 3 in the standard form and find two solutions. Is (, 3) its solution? 41. Sum of the digits of a two digit number is 14. If we add 18 to the original number, the digits interchange their places. Write two equations for these two statements. 8

9 4. The food charges in a hostel are as follows : For the first day, the charges are Rs 100 and for the subsequent days it is Rs 0 per day. Taking the number of days as x and total charges as Rs. y, write a linear equation for this information and draw its graph. 43. Draw the graphs of the equations x + y 10 = 0 and x y + 4 = 0 on the same graph paper. 44. Solve for x : 7x x 1 x = 3 4. Sum of the digits of a two digit number is 1. If 18 is added to the original number the digits interchange their places. Write two linear equations representing these situations. 46. Solve the following equation : x (x ) = Give geometric representation of x + 3 = 0 as an equation (i) in one variable (ii) in two variables. 48. Solve for x : 3 x 7 x + 1 x + = Draw the graph of equation x + 3y = 6. From the graph find the value of x when y = Draw the graph of two lines, whose equations are 3x y = 4 and x + y 3 = 0 on the same graph paper and find the co-ordinates of the point where two lines intersect. 1. Draw the graph of two lines whose equations are 3x y + 6 = 0 and x + y 6 = 0 on the same graph paper. Find the area of the triangle formed by two lines and x-axis. 1. Draw the graph of the linear equation 4x + y = 6. At what points the graph of the equation cuts the x-axis and the y-axis?. Draw the graphs of the equations x + y = 6 and x + 3y = 16 on the same graph paper. Find the co-ordinates of the points where the two lines intersect. 3. Show that the points A (1, ), B ( 1, 16) and C (0, 7) lie on the graph of linear equation y = 9x The force exerted to pull a cart is directly proportional to the acceleration produced in the body. Express the statement as a linear equation of two variables and draw the graph of the same by taking the constant mass equal to 6 kg. Read from the graph, the force required when the LONG ANSWER TYPE QUESTIONS A. Important Questions acceleration produced is (i) m/sec (ii) 6m/ sec.. The linear equation that converts Fahrenheit (F) to Celsius (C) is given by the relation F 160 C = 9 (i) If the temperature is 86 F, what is the temperature in Celsius? (ii) If the temperature is 3 C, what is the temperature in Fahrenheit? (iii) If the temperature is 0 F, what is the temperature in Celsius? (iv) What is the numerical value of the temperature which is same in both the scales? B. Questions From CBSE Examination Papers 1. The taxi fare in a city is a follows : For the first kilometer, the fare is Rs, 8 and for the subsequent distance it is Rs /km. Taking the distance covered as x km and total fare as Rs y, write a linear equation for this information, and draw its graph.. Solve : 4 1 x + x = 6 3. Rs 7 is in the form of 0 paise, paise and 0 paise coins. The number of paise coins is double the number of 0 paise coins but half the number of 0 paise coins. Find the number of coins of each type. 4. Solve : y 3 + y 4 y 1 =

10 . A number consists of digits. The digit at tens place is times the digit in units place. The number formed by reversing the digit is 7 less than the original number. Find the number. 6. Solve for x : x + 3 3x + 1 ( x ) = A man leaves half his property to his wife, one third of the remaining to his son and the rest to his daughter. If daughter s share is Rs 1000, how much money did the man leave? How much money did his wife and son each get? 8. You know that the force applied on a body is directly proportional to the acceleration produced in the body. If constant of proportionality is, write an equation to express this situation and plot the graph of equation. 9. Solve for x : 3 x + 4( x + 1) + = (x + 1) The total monthly expenditure of a household consists of a fixed expenditure on house rent as Rs 00 and the expenditure on rice which is available at Rs 0 per kg. Write a linear equation assuming the consumption of rice to be x kg per month and the total expenditure of the household per month as Rs y. Draw the graph of the equaiton. 11. I am three times as old as my son. Five years later, I shall be two and a half times as old as my son. How old am I and how old is my son? 1. If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as units. Also read from the graph the work done when the distance travelled by the body is units. 13. Solve for x : x 10 4x + + = Draw the graph of the linear equation x 3y + 7 = 0 and hence find the coordinates of the point where the line intersects x-axis. 1. If (, 3) and (4, 0) lie on the graph of equation ax + by = 1, find the values of a and b. Plot the graph of equation obtained. 16. Draw the graphs of the equations x y = 3 and 3x + y = 1 on the same coordinate axes. Also, find the point of intersection of the two lines from the graphs. 17. The following values of x and y are thought to satisfy a linear equation. x 1 y 1 3 Draw the graph, using the values of x and y. At what points the graph cuts the x-axis and y-axis. 18. Draw the graph of equation 3x + y = 6. Also, find the points where the line intersect x-axis and y-axis. 19. Draw the graphs of each of the equations x y 3 = 0 and 4x + 3y 1 = 0 on the same graph years before, age of a mother was 3 times the age of her daughter. Write a linear equation to represent this situation and draw its graph. 1. The parking charges of a car in a parking lot is Rs. 30 for the first two hours and Rs 10 for subsequent hours. Taking total parking time to be x hours and total charges as Rs y, write a linear equation in two variables to express the above statement. Draw a graph for the linear equation and read the charges for five hours.. Draw the graph of the linear equation x + 3y = 1 (i) Write the co-ordinates of a point where graph intersects x-axis. (ii) From the graph show whether points (3, ) and ( 3, 6) are the solution of the equation. 3. A water tank is getting filled up by water flowing at the rate of 1 cm 3 /sec. If the volume of water filled in y seconds is x cm 3, write a linear equation in two variables to represent this situation. Draw a graph for the equation formed and hence find the volume of water filled in 9 seconds. 4. Draw the graph of the equation x + 3y 6 = 0. (i) Using graph paper determine whether x = 3 and y = 0 is a solution (ii) Find the value of y, if x = 3 and (iii) Find the value of x, if y = from the graph and verify.. Solve for x : x 7 3( = x + 1) ( ) = 6. Solve for x : 4 x + x+ 7)

11 7. A and B are friends. A is elder to B by years. B s sister C is half the age of B while A s father D is 8 years older than twice the age of B. If the present age of D is 48 years, find the present ages of A, B and C. 8. Solve for x : 3 x 4x + x = 3 1 FORMATIVE ASSESSMENT Activity Objective : To draw the graph of a linear equation. Materials Required : Graph paper, geometry box, etc. Procedure : Let us draw the graph of the equation x + y = Write the given equation x + y = 11 as y = 11 x. Give some suitable values to x and find the corresponding value of y. When x = 1, then y = = = When x = 3, then y = = = 4 When x = 9, then y = 11 9 = =1 3. Put the corresponding values of x and y in the tabular form as shown below. x y Now plot the points A (1, ), B (3, 4) and C(9, 1) on a graph paper.. Join AB and extend it on both sides to obtain the required graph of the equation x + y =11. Also, check whether the point C(9, 1) lies on the line or not. 6. Pick another point on this line say D(, 3). Check whether it is a solution of the given equation or not. 7. Now take any point not lying on the line AB say E(3, ). Check whether it is a solution of the given equation or not. Observations : 1. After joining AB, if we extend it on both sides, we see that the point C(9, 1) lies on the line AB. It means every point whose coordinates satisfy the given equation lies on the line AB.. The point D(, 3) lies on the line AB. Also, + 3 = + 6 = 11. So, (, 3) is a solution of the given equation. Similarly, the point P(7, ) lies on the line AB. Also, 7 + = = 11. So, (7, ) is also a solution of the given equation. It implies every point (a, b) on the line AB gives a solution x = a, y = b of the given equation. 3. The point E(3, ) does not lie on the line AB. Also, 3 + = = It implies any point which does not lie on the line AB is not a solution of the given equation. Conclusion : From the above activity, we can conclude that : (i) Every point on the line satisfies the equation of the line. (ii) Every solution of the equation is a point on the line. (iii) Any point which does not lie on the line is not a soluton of the equation. Do Yourself : Draw the graphs of each of the following equations : 1. x + y =. x y = x y = 8 4. x 3y = 9 In each case, verify that : (i) Every point on the line satisfies the equation. (ii) Every solution of the equation is a point on the line. (iii) Any point which does not lie on the line is not a solution of the equation. 11