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10th Class Chapter No 3 - PAIR OF LINEAR EQUATIONS IN TWO VARIABLES in Maths for CBSE NCERT

NCERT Solution for Class 10 Math Chapter 3

Linear Equations in Two Variables

 
A linear equation is an algebraic equation where term is constant or product of constant however different variables can occur in different terms. An example of a linear equation with one variable x can be written as : ax + b = 0, where a and b are constants and a ≠ 0. The constants can be numbers, parameters, even non-linear functions of parameters, and the distinction between variables and parameters depend on the problem.Linear equations can also have 1 or more variables. An example of a linear equation with 3 variables, c, d, and e, is given by: cx + dy + ez + f = 0, where c, d, e, and f are constants and c, d, and e are non-zero.
 

NCERT Solution for Class 10 Math-Pair of Linear Equations in Two Variables

Linear equations in 2 variables are equations which can be expressed as cy + dz + e = 0, where c, d and e are real numbers and both c, and d are not zero. The solution of these equations is a pair of values for y and z  which makes both sides of the equation equal.

Let’s have a look at the solutions of linear equations in 2 variables. Consider the equation 2x + 3y = 5. There are two variables in this equations y and z.

Scenario 1

Let’s substitute y = 1 and z = 1 in the Left Hand Side (LHS) of the equation. Hence, 2(1) + 3(1) = 2 + 3 = 5 = RHS (Right Hand Side). Hence, we can conclude that y = 1 and z = 1 is a solution of the equation 2y + 3z = 5. Therefore,y = 1 and z = 1 is a solution of the equation 2y + 3z = 5.
 

Scenario 2

 Let’s substitute y = 1 and z = 7 in the LHS of the equation. Hence, 2(1) + 3(7) = 2 + 21 = 23 ≠ RHS. Therefore, y = 1 and z = 7 is not a solution of the equation 2y + 3z = 5.
 

NCERT Solution for Class 10 Math Chapter 3 Geometrical Representation

 This means that the point (1, 1) lies on the line representing the equation 2y + 3z = 5. Also, the point (1, 7) do not lie on the line. In simple words, every solution of the equation is a point on the line representing it.To generalize, each solution (y, z) of a linear equation in two variables, ay + bz + c = 0, corresponds to a point on the line representing the equation, and vice versa.

Pair of Linear Equations in Two Variables

 
The number of times Rithik eats a mango is half the number of rides he eats an apple and goes to the market ,spends Rs. 20. If a mango costs Rs.3 and an apple costs Rs.4 then how many mangoes and apples did Rithik eat?

Let’s say that the number of apples that Rithik ate is y and the no. of mangoes is x. Now, the situation can be represented as 

y = (½)x … {since he ate mangoes (a) which were half the number of apples (b)}
3a + 4b = 20 … {since each apple (b) costs Rs.4 and mango (a) costs Rs.3}

Both the equation together represent the information about the situation and these 2 linear equations are in the same variables a and b. These are called ‘Pair of Linear Equations in Two Variables’.

To generalize  a pair of linear equations in two variables x and y are

a1x + b1 y + c1 = 0 and a2x + b2 y + c2 = 0.

Where a1, b1, c1, a2, b2, c2 are all real numbers and a12+ b12 ≠ 0, a22+ b22 ≠ 0.

Geometric/Graphical Representation of a Pair of Linear Equations in Two Variables

The geometrical or graphical representation of linear equations in 2 variables is a straight line and a pair of linear equations in two variables will be 2 straight lines which are considered together and also know that when there are two lines in a plane:

  • The two lines will intersect at one point. {Fig.1 (a)}
  • They will not intersect, i.e., they are parallel. {Fig.1 (b)}
  • The two lines will be coincident. {Fig.1 (c)}
linear equations in two variables
 
 

Linear equations can be represented both algebraically and geometrically  and Going back to the earlier example of Rithik let’s represent the situation both algebraically and geometrically.

Solution

 Algebraic Representation

The pair of equations formed is: y = (1/2)x
So, 2y = x
Hence, x – 2y = 0 … (1)
3x + 4y = 20 … (2)

Geometric Representation

To represent these equations graphically we need at least 2 solutions for each equation.These solutions are listed below

x 0 2
y = (1/2)x 0 1

 

x 0 4
y = (20 – 3x)/4 5 2

Now,graph paper and plot the points A(0, 0), B(2, 1) and P(0, 5), Q(4, 2), corresponding to the solutions in tables above. Next, we draw the lines AB and PQ as shown below.

linear equations in two variables

This is the geometric representation of the equations x – 2y = 0 and 3x + 4y = 20.

 
Posted in 10th on February 13 2019 at 03:29 PM

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