1. Given a pair of linear equations:
4x+5y =28, 7x−3y=2
Formulate a word problem for the given system of equations and solve it graphically.
2. To find the condition for consistency and inconsistency for a given set of system of Linear Equations in two variables.
Given a pair of linear equations:
Set I: x+2y− =4 0, x+2y− =6 0 Set II: 2x+4y =10, 3x+6y=12
3. Find whether the following pair of equations are consistent or not by the graphical method. If consistent, solve them.
(a) x+2y=3, 4x+3y=2 (b) 2x +3y=9, 4x +6y =18 (c) x+2y− =4 0, x+2y− =6 0
Group Discussion
Divide the whole class into small groups and ask them to discuss some examples, from daily life where we use the concept of the pair of linear equations in two variables to solve the problems.
The students should write the problems and their corresponding equations.
Multiple Choice Questions
Tick the correct answer for each of the following.
1. A pair of linear equations in two variables cannot have
(a) a unique solution (b) no solution
(c) infinitely many solutions (d) exactly two solutions 2. The pair of equations 3x−2y =5 and 6x− =y 3 have
(a) no solution (b) a unique solution
(c) two solutions (d) infinitely many solutions 3. If a pair of linear equations is inconsistent, then the lines representing them will be
(a) parallel (b) always coincident (c) intersecting or coincident (d) always intersecting
4. If a pair of linear equations has infinitely many solutions, then the lines representing them will be (a) parallel (b) intersecting or coincident
5. The pair of equations 4x−3y+ =5 0 and 8x−6y−10 =0 graphically represents two lines which are (a) coincident (b) parallel
(c) intersecting at exactly one point (d) intersecting at exactly two points 6. The pair of equations y=a and y=b graphically represents lines which are
(a) intersecting at (a, b) (b) intersecting at ( , )b a (c) parallel (d) coincident 7. The pair of equations x = 2 and y = 3 has
(a) one solution (b) two solutions (c) many solutions (d) no solution 8. The value of k for which the pair of equations kx+ =y 3 and 3x+6y=5 has a unique solution is
(a) –1
2 (b) 2 (c) –2 (d) all the above 9. If the lines given by 3x+2ky=2 and 2x+5y+ =1 0 are parallel, then the value of k is
(a) 3 2 (b) 15 4 (c) 2 5 (d) – 5 4
10. One equation of a pair of dependent linear equations is 3x−4y=7. The second equation can be (a) – 6x+8y =14 (b) –6x+8y+14 = 0 (c) 6x+8y=14 (d) −6x−8y−14 =0 11. If x =a and y=b is the solution of the equations x+ =y 5 and x− =y 7, then values of a and b are
respectively
(a) 1 and 4 (b) 6 and –1 (c) – 6 and 1 (d) –1 and –6 12. A pair of linear equations which has a unique solution x = −1, y= −2 is
(a) x− =y 1; 2x+3y=5 (b) 2x−3y=4; x−5y=9 (c) x+ − =y 3 0; x− =y 1 (d) x+ + =y 3 0; 2x−3y+5 = 0
13. Sanya’s age is three times her sister’s age. Five years hence, her age will be twice her sister’s age. The present ages (in years) of Sanya and her sister are respectively
(a) 12 and 4 (b) 15 and 5 (c) 5 and 15 (d) 4 and 12
14. The sum of the digits of a two digit number is 8. If 18 is added to it, the digits of the number get reversed. The number is
(a) 53 (b) 35 (c) 62 (d) 26
15. Divya has only ` 2 and ` 5 coins with her. If the total number of coins that she has is 25 and the amount of money with her is ` 80, then the number of ` 2 and ` 5 coins are , respectively
(a) 15 and 10 (b) 10 and 15 (c) 12 and 10 (d) 13 and 12
Rapid Fire Quiz
State whether the following statements are true (T) or false (F).
1. A linear equation in two variables always has infinitely many solutions.
2. A pair of linear equations in two variables is said to be consistent if it has no solution.
3. A pair of intersecting lines represent a pair of linear equations in two variables having a unique solution.
4. An equation of the form ax + + =by c 0, where a, b and c are real numbers is called a linear equation in two variables.
5. A pair of linear equations in two variables may not have infinitely many solutions. 6. The pair of equations 4x−5y=8 and 8x−10y=3 has a unique solution.
7. A pair of linear equations cannot have exactly two solutions.
Match the Columns
Match the following columns I and II.
Column I Column II
(i) x+ + =y 5 0
5x+2y=–13 (a) infinitely many solutions (ii) 2x+ + =y 7 0 y x− =8 (b) no solution (iii) 3x −4y+ =7 0 8y−6x −14 =0 (c) x =2, y =3 (iv) x+ + =y 1 0 3x−2y=22 (d) x = −1, y= −4 (v) y=5; y=–3 (e) x = −5, y=3 (vi) 3x−2y=0 5x+ =y 13 (f) x =4, y=–5
Class Worksheet
1. Tick the correct answer for each of the following:
(i) The pair of equations 6x −4y+ =9 0 and 3x−2y+10 =0 has (a) a unique solution (b) no solution
(c) exactly two solutions (d) infinitely many solutions (ii) The pair of equations x =a and y=b graphically represents lines which are
(a) coincident (b) parallel (c) intersecting at ( , )a b (d) intersecting at ( , )b a
(iii) If the lines given by 2x −5y+10=0 and kx+15y−30 =0 are coincident, then the value of k is (a) –6 (b) 6 (c) 1
3 (d)
−1 3
(iv) If x =a, y=b is the solution of the equation x+ =y 3 and x − =y 5, then the values of a and b are, respectively
(a) 4 and –1 (b) 1 and 2 (c) –1 and 4 (d) 2 and 3 (v) If we add 1 to the numerator and denominator of a fraction, it becomes 1
2. It becomes 1 3 if we only add 1 to the denominator. The fraction is
(a) 3 4 (b) 2 5 (c) 3 5 (d) 1 4 2. State whether the following statements are true or false. Justify your answer.
(i) The pair of equations 3x−4y=1 and 4x+3y=1 has a unique solution.
(ii) For the pair of equations 4x+λy=– and 63 x+9y+ =4 0 to have no solution, the value of λ should not be 6.
3. (i) If 2x+ =y 23 and 4x− =y 19, find the values of 3y−4x and y
x +3.
(ii) The angles of a cyclic quadrilateral ABCD are
∠ =A (6x +10 ,)° ∠B =( )5x °, ∠C = (x+ °y) , ∠ =D (3y−10)° Find x and y, and hence the value of the four angles.
4. (i) Draw the graphs of the equations y=3, y =5 and 2x− − =y 4 0. Also, find the area of the quadrilateral formed by the lines and the y-axis.
(ii) A motorboat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.
Paper Pen Test
Max. Marks: 25 Time allowed: 45 minutes 1. Tick the correct answer for each of the following:
(i) If a pair of linear equations is consistent, then the lines will be (a) always intersecting (b) always coincident
(c) intersecting or coincident (d) parallel 1 (ii) The pair of equations x+2y− =3 0 and 4x+5y=8 has
(a) no solution (b) infinitely many solutions
(c) a unique solution (d) exactly two solutions 1 (iii) The value of c for which the pair of equations 4x −5y+ =7 0 and 2cx −10y+ =8 0 has no solution is (a) 8 (b) – 8 (c) 4 (d) – 4 1 (iv) A pair of linear equations which has a unique solution x =1, y= −3 is
(a) x− =y 4; 2x+3y=5 (b) 2x− = −y 5; 5x−2y=11
(c) 3x+ =y 0; x+2y= −5 (d) x+ = −y 2; 4x+3y=5 2 (v) Anmol’s age is six times his son’s age. Four years hence, the age of Anmol will be four times his
son’s age. The present age in years, of the father and the son are respectively
(a) 24 and 4 (b) 30 and 5 (c) 36 and 6 (d) 24 and 3 2 2. State whether the following statements are true or false. Justify your answer.
(i) The equations x y
2 1 5 0
+ + = and 4 8 8
5 0
x+ y+ = represent a pair of coincident lines.
(ii) For all real values of k, except –6, the pair of equations kx −3y =5 and 2x+ =y 7 has a unique solution. 2 × 2 = 4 3. (i) For what values of a and b, will the following pair of linear equations have infinitely many
solutions?
x+2y=1; (a b x− ) + +(a b y) = + −a b 2 (ii) Solve for x and y
x a y b a b + = + , x a y b 2+ 2 = 2, a b, ≠0 3 × 2 = 6
4. (i) Graphically solve the pair of equations: 2x+ =y 6, 2x− + =y 2 0
Find the ratio of the areas of the two triangles formed by the lines representing these equations with the x-axis and the lines with the y-axis.
(ii) Saksham travels 360 km to his home partly by train and partly by bus. He takes four and a half hours if he travels 90 km by bus and the remaining by train. If he travels 120 km by bus and remaining by train, he takes 10 minutes longer. Find the speed of the train and the bus separately. 4 × 2 = 8