**NCERT Solution for Class 10 Math Chapter 3**

**Linear Equations in Two Variables**

**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

## 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**

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

a_{1}x + b_{1} y + c_{1} = 0 and a_{2}x + b_{2} y + c_{2} = 0.

Where a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} are all real numbers and a_{1}^{2}+ b_{1}^{2} ≠ 0, a_{2}^{2}+ b_{2}^{2} ≠ 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)}

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