4 Marks Questions

1. Solve the following pair of linear equationsin substitution method(i) 2x − 7y = 3 (ii) 4x + y = 21

Sol: 2x − 7y = 3 .......... (1)

4x + y = 21 .......... (2)From equation (2) y = 21 − 4xSubstituting this in equation (1)2x − 7y = 3⇒ 2x − 7(21 − 4x) = 3 ⇒ 2x − 147 + 28x = 3⇒ 30x = 3 + 147 ⇒ 30x = 150

150x = ⎯⎯ = 5

30Substituting this in equation (2)4x + y = 21⇒ 4(5) + y = 21 ⇒ 20 + y = 21y = 21 − 20 ⇒ y = 1∴ (x, y) = (5, 1)

2. Solve the following pairs of equation byreducing them to a pair of Linear equations.

2 2 1 3 2(i) ⎯ + ⎯ = ⎯, ⎯ + ⎯ = 1

x 3y 6 x ySol: Given equations are

2 2 1⎯ + ⎯ = ⎯ .......... (1) x 3y 6 3 2⎯ + ⎯ = 1 .......... (2)x y

1 1Let ⎯ = a and ⎯ = b

x yEquations (1) and (2) reduces to

2 12a + ⎯ b = ⎯

3 6

12a + 4b = 1 .............. (3)and 3a + 2b = 1 ......... (4)

(3) ⇒ 12a + 4b = 1

(4) × 2 ⇒ 6a + 4b = 2Subtracting, 6a = − 1

−1a = ⎯6

−1Substituting a = ⎯ in equation (4)

6 3a + 2b = 1

−13 (⎯) + 2b = 16

−1⎯ + 2b = 12

1 2 + 1 32b = 1 + ⎯ = ⎯ = ⎯

2 2 2−1 3 ∴ a = ⎯ and b = ⎯

6 41 −1

but a = ⎯ = ⎯ x 6

x = −61 3 4

b = ⎯ = ⎯ ⇒ y = ⎯y 4 3

4(x, y) = (−6, ⎯)3

3. The coach of a cricket team buys 3 batsand 6 balls for Rs.3900. Later he buysanother bat and 3 more balls of the samekind for Rs.1300. Represent this situationalgebraically and graphically.

Sol: Let the price of a bat be Rs.x and that ofa ball be Rs.y. It is given that 3 bats and6 balls are bought for Rs.3900.

3x + 6y = 3900It is also given that one bat and 3 balls ofthe same kind cost Rs.1300.

x + 3y = 1300The Algebraic representation of the givensituation is

3x + 6y = 3900

x + 3y = 1300

Graphical Representation:

Table for 3x + 6y = 3900 is

Table for x + 3y = 1300

From the graph the two lines intersect at thepoint (1300, 0).

email: help@eenadupratibha.netøŒ‰vÚÛî¦ô¢Ù 10 áì÷J 2020

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Pair of Linear Equations in 2 variables

What are consistent equations?

Target-2020

TenthMathematics Paper-1

100100

1. Check whether the following pair ofLinear equations represent parallel linesor not? 6x − 3y + 10 = 0, 2x − y + 9 = 0

Sol: Given equations are 6x − 3y + 10 = 0,2x − y + 9 = 0

Here a1 = 6, b1 = −3, c1 = 10, a2 = 2, b2 = −1, c2 = 9a1 6 b1 −3 c1 10⎯ = ⎯ = 3; ⎯ = ⎯ = 3; ⎯ = ⎯ a2 2 b2 −1 c2 9

a1 b1 c1∴ ⎯ = ⎯ ≠ ⎯a2 b2 c2

∴ They represent parallel lines.2. Airtel company is charging Rs.399 for

1.5GB data at a speed of 5 mbps and100 minutes voice calls. Write a linearequation with two variables for givendata.

Sol: Let the cost of 1GB data = Rs.xThen the cost of 1.5 GB data = Rs.1.5xandLet the cost of 1 minute voice call = Rs.yThen the cost of 100 minute voice calls = Rs.100yGiven sum of these two = Rs.399∴ 1.5x + 100y = 399 or 3⎯ x + 100y = 399 or 3x + 200y = 7982

is the required equation.3. Covert the following into linear

4 3 5 7equations ⎯ + ⎯ = 12; ⎯ − ⎯ = 9x y x y

1Sol: To write linear equations we put ⎯ = a

x1and ⎯ = b then we gety

4a + 3b = 12; 5a − 7b = 9 which are in linear form.

1. For what value of m, the following systemof equations will have a unique solution?3x + my = 10 and 9x + 12y = 30

Sol: Given: 3x + my = 10, 9x + 12y = 30a1 = 3, b1 = m, and a2 = 9, b2 = 12The equations will have unique solution

a1 b1 3 mSo, = ⎯ ≠ ⎯ ⇒ ⎯ ≠ ⎯

a2 b2 9 1212 × 3

m ≠ ⎯⎯ ⇒ m ≠ 49

∴ m ≠ 4 the above system of equationswill have unique solution i.e., R − {4}.

2. Solve the following equations by elimina-tion method(i) 2x + y − 5 = 0 and 3x − 2y − 4 = 0

Sol: 2x + y − 5 = 0 ⇒ 2x + y = 5 ........ (1)3x − 2y − 4 = 0 ⇒ 3x − 2y = 4 ........ (2)(1) × 3 ⇒ 6x + 3y = 15(2) × 2 ⇒ 6x − 4y = 8

⎯⎯Subtracting 7y = 7

⎯⎯7 y = ⎯ = 17

Substituting y = 1 in equation (1)2x + y = 52x + 1 = 52x = 5 − 1 = 4

4x = ⎯ = 2 ⇒ x = 22

Solution x = 2, y = 13. 2 kg brinjal and 4 kg tomato total Rs.120.

After two days 4 kg brinjal and 5 kg toma-to total Rs.160. Express this situation inlinear equation.

Sol: Let, cost of 1 kg brinjal = Rs.xCost of 1 kg tomato = Rs.yCost of 2 kg brinjal and 4 kg tomato = Rs.120∴ 2x + 4y = 120Cost of 4 kg brinjal and 5 kg tomato = Rs.160∴ 4x + 5y = 160

4. Two numbers differ by 3 and their sum is15. Find the numbers.

Sol: Let the two numbers be x and y and x > y.By the sum the two numbers differ by 3then we get x − y = 3 ....... (1)Also given that their sum is 15 means x + y = 15 ....... (2)By adding (1) and (2), we get2x = 18 ⇒ x = 9Substitute the value of ‘x’ in (2), we get9 + y = 15 ⇒ y = 15 − 9 = 6∴ The numbers are x = 9, y = 6

5. Solve the equations 3x – 5y = 4, 3y – 4x = 2

by substitution method.Sol: Given equations are

3x − 5y = 4 ....... (1) and3y − 4x = 2 ⇒ −4x + 3y = 2 ....... (2)

2 + 4xFrom (2) y = ⎯⎯⎯ . Substitute

3 this value in (1), we get

2 + 4x3x − 5 (⎯⎯⎯) = 43

Multiply with 3 on both sides, we get9x − 5 (2 + 4x) = 4 × 3 = 12⇒ 9x − 10 − 20x = 12

22⇒−11x = 12 + 10 = 22 ⇒ x = − ⎯ = − 211

Substitute the value of ‘x’ in (1) we get3(−2) − 5y = 4 ⇒ −6 − 4 = 5y⇒ 5y = −10 ⇒ y = −2Solution is (x, y) = ( −2, −2)

x 1300 0y 0 650

x 1000 100y 100 400

2 Marks Questions

1. For which value of ‘K’ pair of linear equa-tions 3x + 4y + 2 = 0 and 9x + 12y + K = 0are consistent?

2. Check whether the pair of linear equations2x – 3y = 5 and 4x – 6y = 15 are consistent?

3. Two numbers differ by 3 and their productis 54. Find the numbers.

Additional Questions

1 Mark Questions

1. What are consistent equations?A: If a pair of linear equations have unique

solutions then they are called consistent equations.

2. Give an example for consistent equations?

A: Example for consistent equations are2x + y − 5 = 0 and 3x − 2y − 4 = 0 or anysuitable pair of equations

3. A linear equation in two variables has........... solutions.

A: many4. The point of intersection of x + y = 6 and

x − y = 4 is ........... .A: (5, 1)5. The value of x in the equation

3x − (x − 4) = 3x + 1 is ........... .A: 36. The graph y = ax + b is a straight line

which intersects x–axis at ........... .−bA: (⎯ , 0)a

7. The graph of a linear equation in twovariables is a ........... .

A: Straight line8. If ax + by = c and px + qy = r has unique

solution, then ........... .a b

A: ⎯ ≠ ⎯p q9. Sum of two numbers is 44 is represent-

ed by the equation........... .A: x + y = 44

1⎯⎯2 Mark Questions