**Solving Two Variable Equations**

In math if there is two variables in an, equation, we say that there are two unknowns. Two solve an equation with two variables, we need two equations of two variables.

For examples, In 2x + 3y = 10, it has two variables.

To solve this we need to be given one more equations. Otherwise, we can generalize one of the variables value and get the other one corresponding to it.

Now let us see few problems on this topic solving two variables equation.

**Ex 1. Solve the following equations: 3x + 2y = 3**

**
x+ y = 2**

**Sol :** Given: 3x +2y = 3 ---------- (1)

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

(1) – 2 * (2) => 3x + 2y = 3

2x + 2y = 4

We get, x = -1

Therefore (2) => -1 + y = 2 => y = 3

Therefore the solution is (-1, 3)

**Ex 2 : Solve the following equations: y – x = 3 and y = x ^{2} – 12x + 45.**

**Sol :** Given: y – x = 3 -----------(1)

Y = x^{2} – 12x + 45 ----------- (2)

Therefore (1) => y = x + 3 ---------- (3)

By using (3) in (2), (2) => x + 3 = x^{2} – 12x + 45.

`=>` x^{2} -13x + 42 = 0

`=>` (x – 6) (x -7) = 0

`=>` Therefore x = 6,7

Therefore (3) => When x = 6, y = 6+3 = 9

When x = 7, y = 7+3 = 10

Therefore the solutions are (6,9) and (7,10)

**Ex 3: Solve the following equations:**

**x ^{2} + y^{2} = 25 and x – y + 7 = 0**

**Sol :** Given: x^{2} + y^{2} = 25 ----------- (1)

x – y + 7 = 0 --------------(2)

(2) => y = x + 7 -------------(3)

Therefore (1) = x^{2} + (x + 7)^{2} = 25

= x^{2} + x^{2} + 14x + 24 = 0

= 2x^{2} + 14x + 24 = 0

= x^{2} + 7x + 12 = 0

= (x + 3 ) ( x + 4 ) = 0

Therefore x = -3, -4

Therefore (3) = when we = -3, y = -3 + 7 = 4

When x = -4, y = 3

Therefore the solutions are (-3, 4) and (-4, 3).

Between, if you have problem on these topics Definition Dependent Variable, please browse expert math related websites for more help on Define Natural Numbers.

- Solve the following equations:

5x – y = 13 and x – y = 5.

**[Ans: (2, -3)]**

2. Solve the following equations:

x = 3y -1 and x = y^{2} + 2y – 7

**[Ans: (11, 3) and (-2, -4)]**

3. Solve the following equations:

y = x + 1, x^{2} + y^{2} = 5

**[Ans: (1,2) (-2, -1)]**