# RD SHARMA Solutions for Class 10 Maths Chapter 3 - Pairs of Linear Equations in Two Variables

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise 3.114

So, the correct option is (b).

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise 3.115

If am ≠ bl, then the system of equations

ax + by = c

lx + my = n

(a) has a unique solution

(b) has no solution

(c) has infinite many solution^{ }

(d) may or may not have a solution

So, the correct option is (a).

So, the correct option is (b).

So, the correct option is (a).

If 2x-3y=7 and (a+b)x – (a+b-3)y = 4a+b represent coincident lines than a and b satisfy the equation

(a) a+5b=0 (b) 51+b=0 (c) a-5b=0 (d) 5a-b=0

So, the correct option is (c).

If a pair of linear equations in two variables is consistent, then the lines represented by two equations are

(a) Intersecting (b) parallel

(c) always coincident (d) intersecting or coincident

Consistent solution means either linear equations have unique solutions or infinite solutions.

⇒ In case of unique solution; lines are intersecting

⇒ If solutions are infinite, lines are coincident.

So, lines are either intersecting or coincident

So, the correct option is (d).

So, the correct option is (c).

The area of the triangle formed by the lines

y=x, x=6, and y=0 is

(a) 36 sq. units

(b) 18 sq. units

(c) 9 sq. units

(d) 72 sq. units

So, the correct option is (b).

If the system of equations 2x + 3y=5, 4x + ky =10 has infinitely many solutions, then k=

(a) 1

(b)

(c) 3

(d) 6

So, the correct option is (d).

If the system of equations kx - 5y = 2, 6x +2y=7 has no solution, then k=

(a) -10

(b) -5

(c) -6

(d)-15

So, the correct option is (d).

The area of the triangle formed by the lines

x = 3, y = 4 and x = y is

(a) sq. unit

(b) 1 sq. unit

(c) 2 sq. unit

(d) None of these

So, the correct option is (a).

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise 3.116

The area of the triangle formed by the lines

2x + 3y = 12, x – y – 1= 0 and x = 0

(a) 7 sq. units

(b) 7.5 sq. units

(c) 6.5 sq. units

(d) 6 sq. units

So, the correct option is (b).

The sum of the digits of a two digit number is 9. If 27 is added to it, the digits of the number get reversed. The number is

- 25
- 72
- 63
- 36

If x = a, y = b is the solution of the system of equations x - y = 2 and x + y = 4, then the values of a and b are, respectively

- 3 and 1
- 3 and 5
- 5 and 3
- -1 and -3

Since x = a and y = b is the solution of given system of equations x - y = 2 and x + y = 4, we have

a - b = 2 ….(i)

a + b = 4 ….(ii)

Adding (i) and (ii), we have

2a = 6 ⇒ a = 3

⇒ b = 4 - 3 = 1

Hence, correct option is (a).

For what value k, do the equations 3x - y + 8 = 0 and 6x - ky + 16 = 0 represent coincident lines?

- 2
- -2

Aruna has only Rs.1 and Rs.2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is Rs.75, then the number of Rs.1 and Rs.2 coins are, respectively

- 35 and 15
- 35 and 20
- 15 and 35
- 25 and 25

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.1

Seven years ago,

Age of Aftab = x - 7

Age of his daughter = y - 7

According to the given condition,

Three years hence,

Age of Aftab = x + 3

Age of his daughter = y + 3

According to the given condition,

Thus, the given conditions can be algebraically represented as:

x - 7y = -42

x - 3y = 6

Three solutions of this equation can be written in a table as follows:

x | -7 | 0 | 7 |

y | 5 | 6 | 7 |

Three solutions of this equation can be written in a table as follows:

x | 6 | 3 | 0 |

y | 0 | -1 | -2 |

The graphical representation is as follows:

**Concept insight:**In order to represent a given situation mathematically, first see what we need to find out in the problem. Here, Aftab and his daughter's present age needs to be found so, so the ages will be represented by variables x and y. The problem talks about their ages seven years ago and three years from now. Here, the words 'seven years ago' means we have to subtract 7 from their present ages, and 'three years from now' or 'three years hence' means we have to add 3 to their present ages. Remember in order to represent the algebraic equations graphically the solution set of equations must be taken as whole numbers only for the accuracy. Graph of the two linear equations will be represented by a straight line.

The given conditions can be algebraically represented as:

Three solutions of this equation can be written in a table as follows:

x |
50 | 60 | 70 |

y | 60 | 40 | 20 |

Three solutions of this equation can be written in a table as follows:

x |
70 | 80 | 75 |

y | 10 |
-10 |
0 |

The graphical representation is as follows:

**Concept insight:**cost of apples and grapes needs to be found so the cost of 1 kg apples and 1 kg grapes will be taken as the variables. From the given conditions of collective cost of apples and grapes, a pair of linear equations in two variables will be obtained. Then, in order to represent the obtained equations graphically, take the values of variables as whole numbers only. Since these values are large so take the suitable scale.

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.10

A person rowing at the rate of 5 km/h in still water, takes thrice as much time in going 40 km upstream as in going 40 km downstream. Find the speed of stream.

A motor boat 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.

According to the question,

By using equation (1), we obtain:

3x - 10t = 30 ... (3)

Adding equations (2) and (3), we obtain:

x = 50

Substituting the value of x in equation (2), we obtain:

(-2) x (50) + 10t = 20

-100 + 10t = 20

10t = 120

t = 12

From equation (1), we obtain:

d = xt = 50 x 12 = 600

Thus, the distance covered by the train is 600 km.

**Concept insight:**To solve this problem, it is very important to remember the relation . Now, all these three quantities are unknown. So, we will represent these

by three different

variables. By using the given conditions, a pair of equations will be obtained. Mind one thing that the equations obtained will not be linear. But they can be reduced to linear form by using the fact that . Then two linear equations can be formed which can

be solved easily by elimination method.

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.11

ABCD is a cyclic quadrilateral such that A = (4y + 20)^{o}, B = (3y - 5)^{o}, C = (-4x)^{o} and D = (7x + 5)^{o}. Find the four angles.

We know that the sum of the measures of opposite angles in a cyclic quadrilateral is 180°.

A + C = 180

4y + 20 - 4x = 180

-4x + 4y = 160

x - y = -40 ... (1)

Also, B + D = 180

3y - 5 - 7x + 5 = 180

-7x + 3y = 180 ... (2)

Multiplying equation (1) by 3, we obtain:

3x - 3y = -120 ... (3)

Adding equations (2) and (3), we obtain:

-4x = 60

x = -15

Substituting the value of x in equation (1), we obtain:

-15 - y = -40

y = -15 + 40 = 25

A = 4y + 20 = 4(25) + 20 = 120^{o}

B = 3y - 5 = 3(25) - 5 = 70^{o}

C = -4x = -4(-15) = 60^{o}

D = -7x + 5 = -7(-15) + 5 = 110^{o}**Concept insight: **The most important idea to solve this problem is by using the fact that the sum of the measures of opposite angles in a cyclic quadrilateral is 180^{o}. By using this relation, two linear equations can be obtained which can be solved easily by eliminating a suitable variable.

Meena went to a bank to withdraw Rs 2000. She asked the cashier to give her Rs 50 and Rs 100 notes only. Meena got 25 notes in all. Find how many notes Rs 50 and Rs 100 she received.

The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row there would be 2 rows more. Find number of students in the class.

One says, "Give me a hundred, friend! I shall then become twice as rich as you". The other replies, "If you give me ten, I shall be six times as rich as you". Tell me what is the amount of their (respective) capital?

Let the money with the first person and second person be Rs x and Rs y respectively.

According to the question,

x + 100 = 2(y - 100)

x + 100 = 2y - 200

x - 2y = -300 ... (1)

6(x - 10) = (y + 10)

6x - 60 = y + 10

6x - y = 70 ... (2)

Multiplying equation (2) by 2, we obtain:

12x - 2y = 140 ... (3)

Subtracting equation (1) from equation (3), we obtain:

11x = 140 + 300

11x = 440

x = 40

Putting the value of x in equation (1), we obtain:

40 - 2y = -300

40 + 300 = 2y

2y = 340

y = 170

Thus, the two friends had Rs 40 and Rs 170 with them.**Concept insight: **This problem talks about the amount of capital with two friends. So, we will represent them by variables x and y respectively. Now, using the given conditions, a pair of linear equations can be formed which can then be solved easily using elimination method.

A shopkeeper sells a saree at 8% profit and a sweater at 10% discount, thereby getting a sum of Rs.1008. If she had sold the saree at 10% profit and sweater at 8% discount, she would have got Rs.1028. Find the cost price of the saree and the list price (price before discount) of the sweater.

In a competitive examination, one mark is awarded for each correct answer while ½ mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly?

A shopkeeper gives book on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Latika paid Rs.22 for a book kept for 6 days, while Rs.16 for the book kept for four days. Find the fixed charges and charge for each extraday.

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.2

3x - 5y = 20

6x - 10y = - 40

2x + y - 11 = 0

x - y - 1 = 0

x + 2y - 7 = 0

2x - y - 4 = 0

Draw the graphs of the equations 5x - y = 5 and 3x - y = 3. Determine the co-ordinates of the vertices of the triangle formed by these lines and the y axis. Calculate the area of the triangle so formed.

Three solutions of this equation can be written in a table as follows:

x | 0 | 1 | 2 |

y | -5 | 0 | 5 |

x | 0 | 1 | 2 |

y | -3 | 0 | 3 |

The graphical representation of the two lines will be as follows:

It can be observed that the required triangle is ABC.

The coordinates of its vertices are A (1, 0), B (0, -3), C (0, -5).

**Concept insight: **In order to find the coordinates of the vertices of the triangle so formed, find the points where the two lines intersects the y-axis and also where the two lines intersect each other. Here, note that the coordinates of the intersection of lines with y-axis is taken and not with x-axis, this is because the question says to find the triangle formed by the two lines and the y-axis.

Form the pair of linear equations in the following problems, and find their solutions graphically.

(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

(ii) 5 pencils and 7 pens together cost Rs 50, whereas 7 pencils and 5 pens together cost Rs 46. Find the cost of one pencil and that of one pen.

(iii) Champa went to a 'Sale' to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, "The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased". Help her friends to find how many pants and skirts Champa a bought.

(i) Let the number of girls and boys in the class be x and y respectively.

According to the given conditions, we have:

x + y = 10

x - y = 4

x + y = 10 x = 10 - y

Three solutions of this equation can be written in a table as follows:

x | 4 | 5 | 6 |

y | 6 | 5 | 4 |

x - y = 4 x = 4 + y

Three solutions of this equation can be written in a table as follows:

x | 5 | 4 | 3 |

y | 1 | 0 | -1 |

The graphical representation is as follows:

From the graph, it can be observed that the two lines intersect each other at the point (7, 3).

So, x = 7 and y = 3.

Thus, the number of girls and boys in the class are 7 and 3 respectively.

(ii) Let the cost of one pencil and one pen be Rs x and Rs y respectively.

According to the given conditions, we have:

5x + 7y = 50

7x + 5y = 46

Three solutions of this equation can be written in a table as follows:

x | 3 | 10 | -4 |

y | 5 | 0 | 10 |

Three solutions of this equation can be written in a table as follows:

x | 8 | 3 | -2 |

y | -2 | 5 | 12 |

The graphical representation is as follows:

From the graph, it can be observed that the two lines intersect each other at the point (3, 5).

So, x = 3 and y = 5.

Therefore, the cost of one pencil and one pen are Rs 3 and Rs 5 respectively.

(iii)

Let us denote the number of pants by x and the number of skirts by y. Then the equations formed are:

y = 2x - 2 ...(1)

and y = 4x - 4 ...(2)

Let us draw the graphs of Equations (1) and (2) by finding two solutions for each of the equations.

They are given in Table

x | 2 | 0 |

y = 2x - 2 | 2 | -2 |

x | 0 | 1 |

y = 4x - 4 | -4 | 0 |

Plot the points and draw the lines passing through them to represent the equations, as shown in fig.,

The two lines intersect at the point (1,0). So, x = 1, y = 0 is the required solution of the pair of linear equations, i.e., the number of pants she purchased is 1 and she did not buy any skirt.**Concept insight: **Read the question carefully and examine what are the unknowns. Represent the given conditions with the help of equations by taking the unknowns quantities as variables. Also carefully state the variables as whole solution is based on it. On the graph paper, mark the points accurately and neatly using a sharp pencil. Also, take at least three points satisfying the two equations in order to obtain the correct straight line of the equation. Since joining any two points gives a straight line and if one of the points is computed incorrect will give a wrong line and taking third point will give a correct line. The point where the two straight lines will intersect will give the values of the two variables, i.e., the solution of the two linear equations. State the solution point.

Given the linear equation 2x + 3y - 8 = 0, write another linear equation in two variables such that the geometrical representation of the pair so formed is:

(i) Intersecting lines

(ii) parallel lines

(iii) Coincident lines

(i) For the two lines a_{1}x + b_{1}x + c_{1} = 0 and a_{2}x + b_{2}x + c_{2} = 0, to be intersecting, we must have

So, the other linear equation can be 5x + 6y - 16 = 0

(ii) For the two lines a_{1}x + b_{1}x + c_{1} = 0 and a_{2}x + b_{2}x + c_{2} = 0, to be parallel, we must have

So, the other linear equation can be 6x + 9y + 24 = 0,

(iii) For the two lines a_{1}x + b_{1}x + c_{1} = 0 and a_{2}x + b_{2}x + c_{2} = 0 to be coincident, we must have

So, the other linear equation can be 8x + 12y - 32 = 0,

**Concept insight: **In order to answer such type of problems, just remember the conditions for two lines to be intersecting, parallel, and coincident. This problem will have multiple answers as their can be many equations satisfying the required conditions.

Graphically, solve the following 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.

The lines AB and CD intersect at point R(1, 4). Hence, the solution of the given pair of linear equations is x = 1, y = 4.

From R, draw RM ⊥ X-axis and RN ⊥ Y-axis.

Then, from graph, we have

RM = 4 units, RN = 1 unit, AP = 4 units, BQ = 4 units

Determine, graphically, the vertices of the triangle formed by the lines y = x, 3y = x, x + y = 8.

From the graph, the vertices of the triangle AOP formed by the given lines are A(4, 4), O(0, 0) and P(6, 2).

Draw the graph of the equations x = 3, x = 5 and 2x - y - 4 = 0. Also, find the area of the quadrilateral formed by the lines and the x-axis.

The graph of x = 3 is a straight line parallel to Y-axis at a distance of 3 units to the right of Y-axis.

The graph of x = 5 is a straight line parallel to Y-axis at a distance of 5 units to the right of Y-axis.

Draw the graphs of the lines x = -2, and y = 3. Write the vertices of the figure formed by these lines, the x-axis and the y-axis. Also, find the area of the figure.

The graph of x = -2 is a straight line parallel to Y-axis at a distance of 2 units to the left of Y-axis.

The graph of y = 3 is a straight line parallel to X-axis at a distance of 3 units above X-axis.

Draw the graphs of the pair of linear equations x - y + 2 = 0 and 4x - y - 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.3

Solve the following systems of equation:

Solve the pair of equations:

Solve the following systems of equation:

21x + 47y = 110

47x + 21y = 162

If x + 1 is a factor of 2x^{3} + ax^{2}
+ 2bx + 1, the find the values of a and b given that
2a - 3b = 4.

Find the solution of the pair of equations and . Hence, find λ, if y = λx + 5.

Find the values of x and y in the following rectangle.

Write an equation of a line passing through the point representing solution of the pair of linear equations x + y = 2 and 2x - y = 1. How many such lines can we find?

Write a pair of linear equations which has the unique solution x = -1, y = 3. How many such pairs can you write?

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.4

Solve each of the following systems of equations by the method of cross-multiplication:

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.5

Determine whether the following system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

x - 3y = 3

3x - 9y = 2

Determine whether the following system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

2x + y = 5

4x + 2y = 10

Determine whether the following system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

3x - 5y = 20

6x - 10y = 40

x - 2y = 8

5x - 10y = 10

Find the value of k for which the following system of equations has a unique solution:

kx + 2y = 5

3x + y = 1

4x + ky + 8 = 0

2x + 2y + 2 = 0

Find the value of k for which the following system of equations has a unique solution:

4x - 5y = k

2x - 3y = 12

Find the value of k for which the following system of equations has a unique solution:

x + 2y = 3

5x + ky + 7 = 0

Find the value of k for which the following systems of equations have infinitely many solutions:

2x + 3y - 5 = 0

6x + ky - 15 = 0

Find the value of k for which the following systems of equations have infinitely many solutions:

4x + 5y = 3

kx + 15y = 9

Find the value of k for which the following systems of equations have infinitely many solutions:

kx - 2y + 6 = 0

4x - 3y + 9 = 0

Find the value of k for which the following systems of equations have infinitely many solutions:

8x + 5y = 9

kx + 10y = 18

Find the value of k for which the following systems of equations have infinitely many solutions:

2x - 3y = 7

(k + 2)x - (2k + 1)y = 3(2k - 1)

Find the value of k for which the following systems of equations have infinitely many solutions:

2x + 3y = 2

(k + 2)x + (2k + 1)y = 2(k - 1)

Find the value of k for which the following systems of equations have infinitely many solutions:

x + (k + 1)y = 4

(k + 1)x + 9y = (5k + 2)

Find the value of k for which the following systems of equations have infinitely many solutions:

kx + 3y = 2k + 1

2(k + 1)x + 9y = 7k + 1

Find the value of k for which the following systems of equations have infinitely many solutions:

2x + (k - 2)y = k

6x + (2k - 1)y = 2k + 5

Find the value of k for which the following systems of equations have infinitely many solutions:

2x + 3y = 7

(k + 1)x + (2k - 1)y = 4k + 1

Find the value of k for which the following systems of equations have infinitely many solutions:

2x + 3y = k

(k - 1)x + (k + 2)y = 3k

kx - 5y = 2

6x + 2y = 7

Find he value of k for which of the following system of equation has no solution:

kx + 3y = k - 3

12x + ky = 6

x + 2y + 7 = 0

2x + ky + 14 = 0

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y = 7

(a - b) x + (a + b)y = 3a + b - 2

Find the values of a and b for which the following system of equations has infinitely many solutions:

x + 2y = 1

(a - b)x + (a + b)y = a + b - 2

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y = 7

2ax + ay = 28 - by

For which value(s) of λ,
do the pair of linear equations λx + y = λ^{2}
and x + λy = 1 have no solution?

For which value(s) of λ,
do the pair of linear equations λx + y = λ^{2}
and x + λy = 1 have infinitely many
solutions?

For which value(s) of λ,
do the pair of linear equations λx + y = λ^{2}
and x + λy = 1 have a unique solution?

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.6

Jamila sold a table and a chair for Rs.1050, thereby making a profit of 10% on a table and 25% on the chair. If she had taken profit of 25% on the table and 10% on the chair she would have got Rs.1065. Find the cost price of each.

Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. She received Rs.1860 as annual interest. However, had she interchanged the amount of investment in the two schemes, she would have received Rs.20 more as annual interest. How much money did she invest in each scheme?

The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, he buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs. 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

The cost of 4 pens and 4 pencils boxes is Rs.100. Three times the cost of a pen is Rs.15 more than the cost of a pencil box. Form the pair of linear equations for the above situation. Find the cost of a pen and a pencil box.

One says, "Give me a hundred, friend! I shall then become twice as rich as you". The other replies, "If you give me ten, I shall be six times as rich as you". Tell me what is the amount of their (respective) capital?

A and B each have a certain number of mangoes. A says to B, "if you give 30 of your mangoes, I will have twice as many as left with you. "B replies, "if you give me 10, I will have thrice as many as left with you. "How many mangoes does each have?

Vijay had some bananas, and he divided them into two lots A and B. He sold first lot at the rate of Rs.2 for 3 bananas and the second lot at the rate of Rs.1 per banana and got a total of Rs.400. If he had sold the first lot at the rate of Rs.1 per banana and the second lot at the rate of Rs.4 per five bananas, his total collection would have been Rs.460. Find the total number of bananas he had.

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.7

Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers.

A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.8

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.

## Chapter 3 - Pairs of Linear Equations in Two Variables Exercise Ex. 3.9

Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

The ages of two friends Ani and Biju differ by 3 years. Ani's father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differs by 30 years. Find the ages of Ani and Biju.

The difference between the ages of Ani and Biju is given as 3 years. So, either Biju is 3 years older than Ani or Ani is 3 years older than Biju.

Let the age of Ani and Biju be x years and y years respectively.

Age of Dharam = 2 × x = 2x years

x - y = 3 ... (1)

Age of Biju = 19 - 3 = 16 years

Case II: Biju is older than Ani by 3 years

y - x = 3 ... (3)

4x - y = 60 ... (4)

Adding (3) and (4), we obtain:

3x = 63

x = 21

Age of Ani = 21 years

Age of Biju = 21 + 3 = 24 years**Concept Insight: **In this problem, ages of Ani and Biju are the unknown quantities. So, we represent them by variables x and y. Now, note that here it is given that the ages of Ani and Biju differ by 3 years. So, it is not mentioned that which one is older. So, the most important point in this question is to consider both cases Ani is older than Biju and Biju is older than Ani. For second condition the relation on the ages of Dharam and Cathy can be implemented . Pair of linear equations can be solved using a suitable algebraic method.

Two years ago, Salim was thrice as old as his daughter and six years later, he will be four years older than twice her age. How old are they now?

The age of the father is twice the sum of the ages of his two children. After 20 years, his age will be equal to the sum of the ages of his children. Find the age of the father.

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