NCERT Solutions for Class 10 Maths Chapter 3 Exercise 3.2 - Pair of Linear Equations in two variables

Q1(i) Solve the following pair of linear equations by the substitution method.

$x+y=14$

$x−y=4$

Answer:

Given two equations,

$x+y=14$.......(1)

$x−y=4$........(2)

Now, from (1), we have

$y=14−x$........(3)

Substituting this in (2), we get

$x−(14−x)=4$

$\Rightarrow x−14+x=4$

$\Rightarrow 2x=4+14=18$

$\Rightarrow x=9$

Substituting this value of x in (3)

$\Rightarrow y=14−x=14−9=5$

Hence, the solution of the given equations is x = 9 and y = 5.

Q1(ii) Solve the following pair of linear equations by the substitution method

$s−t=3$

$\frac{s}{3}+\frac{t}{2}=6$

Answer:

Given two equations,

$s−t=3$..........(1)

$\frac{s}{3}+\frac{t}{2}=6$....... (2)

Now, from (1), we have

$s=t+3$........(3)

Substituting this in (2), we get

$\frac{t+3}{3}+\frac{t}{2}=6$

$\Rightarrow 2t+6+3t=36$

$\Rightarrow 5t+6=36$

$\Rightarrow 5t=30$

$\Rightarrow t=6$

Substituting this value of t in (3)

$\Rightarrow s=t+3=6+3=9$

Hence, the solution of the given equations is s = 9 and t = 6.

Q1(iii) Solve the following pair of linear equations by the substitution method.

$3x−y=39$

$9x−3y=9$

Answer:

Given two equations,

$3x−y=3$......(1)

$9x−3y=9$.....(2)

Now, from (1), we have

$y=3x−3$........(3)

Substituting this in (2), we get

$9x−3(3x−3)=9$

$\Rightarrow 9x−9x+9=9$

$\Rightarrow 9=9$

This is always true, and hence this pair of equations has infinite solutions.

As we have

$y=3x−3$,

One of many possible solutions is x = 1, and y = 0.

Q1(iv) Solve the following pair of linear equations by the substitution method.

$0.2x+0.3y=1.3$

$0.4x+0.5y=2.3$

Answer:

Given two equations,

$0.2x+0.3y=1.3$

$0.4x+0.5y=2.3$

Now, from (1), we have

$y=\frac{1.3−0.2x}{0.3}$........(3)

Substituting this in (2), we get

$0.4x+0.5\frac{1.3−0.2x}{0.3}=2.3$

$\Rightarrow 0.12x+0.65−0.1x=0.69$

$\Rightarrow 0.02x=0.69−0.65=0.04$

$\Rightarrow x=2$

Substituting this value of x in (3)

$\Rightarrow y=\frac{1.3−0.2x}{0.3} =\frac{1.3−0.2 × 2}{0.3} =3$

Hence, the solution of the given equations is

x = 2 and y = 3.

Q1(v) Solve the following pair of linear equations by the substitution method.

$\sqrt 2x+ \sqrt 3y=0$

$\sqrt 3x−\sqrt 8y=0$

Answer:

Given two equations,

$\sqrt 2x+ \sqrt 3y=0$

$\sqrt 3x−\sqrt 8y=0$

Now, from (1), we have

$x=\frac{−\sqrt 3y}{\sqrt 2}$........(3)

Substituting this in (2), we get

$\sqrt 3 (\frac{−\sqrt 3y}{\sqrt 2}) - \sqrt 8y=0$

$\Rightarrow \frac{−\sqrt 3y}{\sqrt 2} - 2\sqrt 2y = 0$

$\Rightarrow y(\frac{−\sqrt 3}{\sqrt 2} - 2\sqrt 2) = 0$

$\Rightarrow y=0$

Substituting this value of y in (3)

$\Rightarrow x = 0$

Hence, the solution of the given equations is,

x = 0, and y = 0 .

Q1(vi) Solve the following pair of linear equations by the substitution method.

$\frac{3x}{2}−\frac{5y}{3} =−2$

$\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$

Answer:

Given,

$\frac{3x}{2}−\frac{5y}{3} =−2$ ....... (1)

$\frac{x}{3}+\frac{y}{2}=\frac{13}{6}$ ........ (2)

From (1) we have,

$x=\frac{(-12 + 10y)}{9}$........(3)

Putting this in (2), we get,

$\frac{\frac{(-12 + 10y)}{9}}{3}+ \frac{y}{2} = \frac{13}{6}$

$\frac{(-12 + 10y)}{27}+ \frac{y}{2} = \frac{13}{6}$

$\frac{(-24 + 20y + 27y)}{54} = \frac{13}{6}$

$47y=141$

$y=3$

Putting this value in (3), we get,

$x=\frac{(-12 + 10× 3)}{9}$

$x =2$

Hence, x = 2 and y = 3.

Q2 Solve 2x + 3y = 11 and 2x − 4y = −24 and hence find the value of ‘m’ for which y = mx + 3.

Answer:

Given two equations,

$2x+3y=11$......(1)

$2x−4y=−24$.......(2)

Now, from (1), we have

$y=\frac{11−2x}{3}$........(3)

Substituting this in (2), we get

$2x−4(\frac{11−2x}{3})=−24$

$\Rightarrow 6x−44+8x=−72$

$\Rightarrow 14x=44−72$

$\Rightarrow 14x=−28$

$\Rightarrow x=−2$

Substituting this value of x in (3)

$\Rightarrow y=\frac{11−2x}{3}=\frac{11−2×(−2)}{3}=\frac{15}{3}=5$

Hence, the solution of the given equations is,

x = −2, and y = 5.

Now,

As it satisfies $y=mx+3$,

$\Rightarrow 5=m(−2)+3$

$\Rightarrow 2m=3−5$

$\Rightarrow 2m=−2$

$\Rightarrow m=−1$

Hence, the value of m is -1.

Q3(i) Form the pair of linear equations for the following problem and find their solution by the substitution method.

The difference between the two numbers is 26, and one number is three times the other. Find them.

Answer:

Let two numbers be x and y, and the bigger number is y.

Now, according to the question,

$y−x=26$......(1)

And

$y=3x$......(2)

Now, substituting the value of y from (2) in (1), we get,

$3x−x=26$

$\Rightarrow 2x=26$

$\Rightarrow x=13$

Substituting this in (2)

$\Rightarrow y=3x=3(13)=39$

Hence, the two numbers are 13 and 39.

Q3(ii) Form the pair of linear equations for the following problem and find their solution by the substitution method

The larger of the two supplementary angles exceeds the smaller by 18 degrees. Find them.

Answer:

Let the larger angle be x and the smaller angle be y

Now, as we know, the sum of supplementary angles is 180. so,

$x+y=180^0$.......(1)

Also given in the question,

$x−y=18^0$.......(2)

Now, from (2) we have,

$y=x−18^0$.......(3)

Substituting this value in (1)

$x+x−18^0=180^0$

$\Rightarrow 2x=180^0+18^0$

$\Rightarrow 2x=198^0$

$\Rightarrow x=99^0$

Now, substituting this value of x in (3), we get

$\Rightarrow y=x−18^0=99^0−18^0=81^0$

Hence, the two supplementary angles are

$99^0$ and $81^0$.

Q3 Form the pair of linear equations for the following problems and find their solution by the substitution method. (iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

Answer:

Let the cost of 1 bat is x and the cost of 1 ball is y.

Now, according to the question,

$7x+6y=3800$......(1)

$3x+5y=1750$......(2)

Now, from (1) we have

$y=\frac{(3800−7x)}{6}$........(3)

Substituting this value of y in (2)

$3x+5\frac{(3800−7x)}{6}=1750$

$\Rightarrow 18x+19000−35x=1750×6$

$\Rightarrow −17x=10500−19000$

$\Rightarrow −17x=−8500$

$\Rightarrow x=\frac{8500}{17}$

$\Rightarrow x=500$

Now, Substituting this value of x in (3)

$y=\frac{(3800−7x)}{6}$

$y =\frac{3800−7×500}{6}$

$y =\frac{3800−3500}{6}=50$

Hence, the cost of one bat is 500 Rs and the cost of one ball is 50 Rs.

Q3. Form the pair of linear equations for the following problems and find their solution by the substitution method. (iv) The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of 10 km, the charge paid is Rs 105, and for a journey of 15 km, the charge paid is Rs 155. What are the fixed charges and the charge per km? How much does a person have to pay for travelling a distance of 25 km?

Answer:

Let the fixed charge is x and the per km charge is y.

Now, according to the question

$x+10y=105$.......(1)

And,

$x+15y=155$.......(2)

Now, from (1) we have,

$x=105−10y$........(3)

Substituting this value of x in (2), we have

$105−10y+15y=155$

$\Rightarrow 5y=155−105$

$\Rightarrow 5y=50$

$\Rightarrow y=10$

Now, substituting this value in (3)

$x=105−10y$

$=105−10(10)$

$=105−100=5$

Hence, the fixed charge is 5 Rs and the per km charge is 10 Rs.

Now, Fair for 25 km :

$\Rightarrow x+25y=5+25(10) $

$=5+250=255$

Hence, fair for 25km is 255 Rs.

Q3.Form the pair of linear equations for the following problems and find their solution by the substitution method. (v) A fraction becomes 911, if 2 is added to both the numerator and the denominator. If 3 is added to both the numerator and the denominator, it becomes 56. Find the fraction.

Answer:

Let the numerator of the fraction be x, and the denominator of the fraction be y

Now, according to the question,

$\frac{x+2}{y+2}=\frac{9}{11}$

$\Rightarrow 11(x+2)=9(y+2)$

$\Rightarrow 11x+22=9y+18$

$\Rightarrow 11x−9y=−4$.........(1)

Also,

$\frac{x+3}{y+3}=\frac{5}{6}$

$\Rightarrow 6(x+3)=5(y+3)$

$\Rightarrow 6x+18=5y+15$

$\Rightarrow 6x−5y=−3$...........(2)

Now, from (1) we have

$y=\frac{11x+4}{9}$.............(3)

Substituting this value of y in (2)

$6x−5(\frac{11x+4}{9})=−3$

$\Rightarrow 54x−55x−20=−27$

$\Rightarrow −x=20−27$

$\Rightarrow x=7$

Substituting this value of x in (3)

$y={11x+4}{9}$

$=\frac{11(7)+4}{9}=\frac{81}{9}=9$

Hence, the required fraction is

$x =\frac{7}{9}$

Q3 Form the pair of linear equations for the following problems and find their solution by the substitution method. (vi) Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

Answer:

Let x be the age of Jacob and y be the age of Jacob's son.

Now, according to the question

$x+5=3(y+5)$

$\Rightarrow x+5=3y+15$

$\Rightarrow x−3y=10$........(1)

Also,

$x−5=7(y−5)$

$\Rightarrow x−5=7y−35$

$\Rightarrow x−7y=−30$.........(2)

Now,

From (1) we have,

$x=10+3y$...........(3)

Substituting this value of x in (2)

$10+3y−7y=−30$

$\Rightarrow −4y=−30−10$

$\Rightarrow 4y=40$

$\Rightarrow y=10$

Substituting this value of y in (3),

$x=10+3y=10+3(10)=10+30=40$

Hence, the present age of Jacob is 40 years and the present age of Jacob's son is 10 years.