## Linear Line Equations

**Introduction:**

Linear line equations have one or additional variables. Linear line equations arises with great consistency in functional mathematics, while they arises quite logically when modeling many incident, they are fundamentally helpful since many non-linear equations should be summarized to linear line equations. A common form of a linear line equations in the two variables x and y is

**y = mx + b**is the general form of linear line equation, where the constant of

**m**is the slope, and

**b**is y-intercept.

Solving Linear Line Equations by Elimination Method:

**Example 1:**Solve

**x – 6y + 3 = 0 and 2x + 2y + 5 = 0**

**.**

**Solution: x – 6y + 3 = 0 → (a)**

**2x + 2y + 6 = 0 → (b)**

Multiply

**(b)**by

**3,**

**(b) => 6x + 6y + 18 = 0 →(c)**

Add

**(a)**and

**(c),**

**=> 7x + 21 = 0**

**=> 7x = – 21**

**=> x = – 3**

Plug

**x = – 1**in

**(b)**

**(b) => 6x + y + 5 = 0**

**=> 6(–1) + y + 5 = 0**

**=> – 6 + y + 5 = 0**

**=> y – 1 = 0**

**=> y = 1**

Therefore the solutions are

**x = – 3**and

**y = 1.**

**Example 2:**Solve the linear line equations given below

**4x – 7y = 8 → (a)**

**2x + 3y = −3 → (b)**

**Solution:**

**Multiply**

**(2)**by

**2**,

**(b) × 2 => 4x + 6y = −6 → (c)**

**Subtract**

**(c)**from

**(a)**

**,**

**(a) => 4x −7y = 8**

**(c) => 4x + 6y = −6**

**=> −y =2**

**=> y = −2**

**Plug value of**

**y**in

**(a),**

**(a) => 4x – 7(−2) = 8**

**=> 4x +14 = 8**

**=> 4x = 8 – 14**

**=> 4x = −6**

**=> x = −6/4 = –3/2**

Hence, the solutions are,

**x = −3/2**and

**y = −2.**

Solving Linear Line Equations by the Substitution Method:

**Example 1:**Solve

**2x + 2y − 6 = 0**and

**3x + y + 4 = 0.**

**Solution:**

**2x + 2y − 6 = 0 → (a)**

**3x + y + 4 = 0**

**→ (b)**

Consider,

**(a) => 2x + 2y − 6 = 0**

**=> 2y =**

**− 2x +**

**6**

**=> y =**

**−x**

**+ 3**

**→ (c)**

Plug

**(c)**in

**(b).**

**(b) =>**

**3x + y + 4 = 0**

**=> 3x + (**

**−x**

**+ 3**

**) + 4 = 0**

**=> 3x**

**− x**

**+ 3 + 6 = 0**

**=> 2**

**x**

**+ 9 = 0**

**=>**

**2x =**

**−**

**18**

**=> x =**

**−**

**9**

Plug

**x =**

**−**

**9**in

**(a).**

**(a) =>**

**2x + 2y − 6 = 0**

**=> 2(**

**−**

**9**

**) + 2y**

**− 6 = 0**

**=>**

**− 18**

**+ 2y − 6 = 0**

**=> 2y**

**−**

**22 = 0**

**=> y =**

**11**

**.**

The solutions are,

**x =**

**−**

**9**and

**y = 11**

**.**

**Example 2:**Solve the linear line equations given below,

**x + 4y = 5 → (a)**

**6x − 2y = 4 → (b)**

**Solution:**

**Consider**

**(b),**

**(b) => 3x – y = 2**

**=> 2y = 6x – 4**

**=> y = 3x**

**–**

**4 → (c)**

**P**lug

**y**in

**(a),**

**(a) => x + 4(3x − 2) = 5**

**=> x + 12x – 8 = 5**

**=>13 x = 13**

**=> x = 1**

**P**lug

**x**in

**(c),**

**(c) => y = 3(1) – 2**

**=> y = 3 – 2**

**=> y = 1**

Hence, the solution are,

**x = 1**and

**y = 1.**