What are consistent equations? - eenadupratibha.net · Subtracting 7y = 7 ⎯⎯ 7 y = ⎯ = 1 7...
Transcript of What are consistent equations? - eenadupratibha.net · Subtracting 7y = 7 ⎯⎯ 7 y = ⎯ = 1 7...
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: [email protected]øŒ‰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