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Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10 PDF Download

Exercise 3.1


Q.1. Akhila went to a fair in her village. She wanted to enjoy rides on the Giant Wheel and play Hoopla (a game in which you throw a rig on the items kept in the stall, and if the ring covers any object completely you get it). The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. Each ride costs Rs 3, and a game of Hoopla costs Rs 4. If she spent Rs 20 in the fair, represent this situation algebraically and graphically. Sol: The pair of equations formed is:
Solution.
The pair of equations formed is:
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
i.e., x - 2y = 0 ....(1)
3x + 4y = 20 ....(2)
Let us represent these equations graphically. For this, we need at least two solutions for each equation. We give these solutions in Table
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Recall from Class IX that there are infinitely many solutions of each linear equation. So each of you choose any two values, which may not be the ones we have chosen. Can you guess why we have chosen x =O in the first equation and in the second equation? When one of the variables is zero, the equation reduces to a linear equation is one variable, which can be solved easily. For instance, putting x =O in Equation (2), we get 4y = 20 i.e.,
y = 5. Similarly, putting y =O in Equation (2), we get 3x = 20 ..,Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10But asPair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10is not an integer, it will not be easy to plot exactly on the graph paper. So, we choose y = 2 which gives x = 4, an integral value.
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Plot the points A (O,O) , B (2,1) and P (O,5) , Q (412) , corresponding to the draw the lines AB and PQ, representing the equations x - 2 y = O and 3x + 4y= 20, as shown in figure
In fig., observe that the two lines representing the two equations are intersecting at the point (4,2),

Q.2. Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” Is not this interesting? Represent this situation algebraically and graphically.
Sol: Let the present age of Aftab and his daughter be x and y respectively. Seven years ago.
Age of Ahab = x - 7
Age of his daughter y - 7
According to the given condition.
(x - 7) = 7(y - 7)
⇒ x - 7 = 7y - 49
⇒ x - 7y = -42
Three years hence
Age of Aftab = x + 3
Age of his daughter = y + 3
According to the given condition,
(x + 3) = 3 (y + 3)
⇒ x+3 = 3y +9
⇒ x - 3y = 6
Thus, the given condition can be algebraically represented as
x - 7y = - 42
x - 3y = 6
x - 7y = - 42 ⇒ x = -42 + 7y
Three solution of this equation can be written in a table as follows:
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
x - 3y = 6 ⇒ x = 6+3y
Three solution of this equation can be written in a table as follows:
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
The graphical representation is as follows:
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
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 daughters present age needs to be found so, so the ages will be represented by variables z 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.

Q.3. The path of a train A is given by the equation 3x + 4y - 12 = 0 and the path of another train B is given by the equation 6x + 8y - 48 = 0. Represent this situation graphically.
Sol:
The paths of two trains are giver by the following pair of linear equations.
3x + 4 y -12 = 0    ...(1)
6x + 8 y - 48 = 0    ... (2)
In order to represent the above pair of linear equations graphically. We need two points on the line representing each equation. That is, we find two solutions of each equation as given below:
We have,
3x + 4 y -12 = 0
Putting y = 0, we get
3x + 4 x 0 - 12 = 0
⇒ 3x = 12
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Putting x = 0, we get
3 x 0 + 4 y -12 = 0
 ⇒ 4y = 12
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Thus, two solution of equation 3x + 4y - 12 = 0 are ( 0, 3) and ( 4, 0 )
We have,
6x + 8y -48 = 0
Putting x = 0, we get
6 x 0 + 8 y - 48 = 0
⇒ 8y = 48
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
⇒ y = 6
Putting y = 0, we get
6x + 8 x 0 = 48 = 0
⇒ 6x = 48
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Thus, two solution of equation 6 x + 8y - 48= 0 are ( 0, 6 ) and (8, 0 )
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Clearly, two lines intersect at ( -1, 2 )
Hence, x = -1,y = 2 is the solution of the given system of equations.

Q.4. Gloria is walking along the path joining (— 2, 3) and (2, — 2), while Suresh is walking along the path joining (0, 5) and (4, 0). Represent this situation graphically.
Sol:
It is given that Gloria is walking along the path Joining (-2,3) and (2, -2), while Suresh is walking along the path joining (0,5) and (4,0).
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
We observe that the lines are parallel and they do not intersect anywhere.

Q.5. On comparing the ratios Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide:
(i) 5x- 4y + 8 = 0
7x + 6y - 9 = 0
(ii) 9x + 3y + 12 = 0
18x + 6y + 24 = 0
(iii) 6x - 3y + 10 = 0
2x - y + 9 = 0

Sol:
We have,
5x - 4 y + 8 = 0
7 x + 6 y - 9 = 0
Here,
a= 5, b1 = -4, c1 = 8
a2 = 7, b2 = 6, c2 = -9
We have,
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
∴ Two lines are intersecting with each other at a point.
We have,
9 x + 3 y +12 = 0
18 + 6 y + 24 = 0
Here,
a1 = 9, b1 = 3, c1 = 12
a2 = 18, b2 = 6, c2 = 24
Now,
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
And Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
∴ Both the lines coincide.
We have,
6 x - 3 y +10 = 0
2 x - y + 9 = 0
Here,
a1 = 6, b= -3, c1 = 10
a2 = 2, b2 = -1, c2 = 9
Now,
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
And Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
∴ The lines are parallel

Q.6. 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.

Sol:
We have,
2x + 3 y - 8 = 0
Let another equation of line is
4x + 9 y - 4 = 0
Here,
a1 = 2, b1 = 3, c1 = -8
a= 4, b2 = 9, c2 = -4
Now,
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
And Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
∴ 2x + 3 y - 8 = 0 and 4 x + 9 y - 4 = 0 intersect each other at one point.
Hence, required equation of line is 4 x + 9y - 4 = 0
We have,
2x + 3y -8 = 0
Let another equation of line is:
4x +6y -4 = 0
Here,
a1 = 2, b1 = 3, c1 = -8
a2 = 4, b2 = 6, c2 = -4
Now,
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
And Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
∴ Lines are parallel to each other.
Hence, required equation of line is 4 x + 6y - 4 = 0.

Q.7. The cost of 2kg of apples and 1 kg of grapes on a day was found to be Rs 160. After a month, the cost of 4kg of apples and 2kg of grapes is Rs 300. Represent the situation algebraically and geometrically.
Sol:
Let the cost of 1 kg of apples and 1 kg grapes be Rs x and Rs y.
The given conditions can be algebraically represented as:
2 x + y = 160 4 x + 2 y = 300
2x + y = 160 ⇒ y = 160 - 2x
Three solutions of this equation cab be written in a table as follows:
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
4x + 2y = 300 ⇒ y = Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Three solutions of this equation cab be written in a table as follows:
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
The graphical representation is as follows:
Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10
Concept insight: cost of apples and grapes needs to be found so the cost of 1 kg apples and 1kg grapes will be taken as the variables from the given condition 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.

The document Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
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FAQs on Pair of Linear Equations in Two Variables - 1 RD Sharma Solutions - Mathematics (Maths) Class 10

1. What are linear equations in two variables?
Ans. Linear equations in two variables are equations that involve two variables and have the highest power of the variables as 1. They can be represented in the form ax + by = c, where a, b, and c are constants, and x and y are variables.
2. How do we solve a pair of linear equations in two variables?
Ans. To solve a pair of linear equations in two variables, we can use different methods such as the substitution method, the elimination method, or the graphical method. These methods involve manipulating the equations to eliminate one variable and find the values of the other variable.
3. What is the substitution method for solving linear equations in two variables?
Ans. The substitution method involves solving one equation for one variable and substituting it into the other equation. By substituting the value, we can solve for the other variable. This method is useful when one equation is already solved for a variable.
4. How does the elimination method work for solving linear equations in two variables?
Ans. The elimination method involves adding or subtracting the equations in such a way that one variable gets eliminated. This leads to a new equation involving only one variable, which can be easily solved. The solution is then substituted back into one of the original equations to find the value of the other variable.
5. What is the graphical method for solving linear equations in two variables?
Ans. The graphical method involves plotting the equations on a coordinate plane and finding the point of intersection. The coordinates of the intersection point represent the solution to the pair of linear equations. This method is useful when the equations are in slope-intercept form (y = mx + b) and can be easily graphed.
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