**Introduction to Linear Equation**

Linear equation is an equation that “**plots a straight line on graphs”**.

**Standard form of Linear equation is:**

**Ax + By = C**

Where A, B, C are the numbers (Coefficients) and x and y are the variables.

**Example**

**Ax + By = C**

Just put numbers in above equation by replacing A, B and C.

- Y = 2x + 1.
- 2x + 3y=7.
- 5x = 6 +3y.

If x increases, y increases double as fast.

**Steps for solving linear equations **

- Expands the brackets.
- Change the order of terms so that all variables are on L.H.S of equations and all constants are on R.H.S of equations.
- Combine like terms
- Solve like terms
- Find the solution and write the answer

**Example**

Linear equation graph for y = 2 x + 1.

**Example**

**Draw a Graph of a linear equation x-3y=3.**

**Step1:**

X- 3y = 3** //equation 1.**

⇒**For solving y we assume that x=0.**

**Putting x=0 in equation 1.**

0 – 3y=3

⇒-3y=3

⇒Y = – 1

**Step 2:**

**For solving x we assume that y=0**

X – 3 (0) = 3

X = 3.

**Step 3**: draw a straight line between x=3 and y=-1

Graph between x and y (3, 0) and (0,-1)

Draw a straight line through it.

**Example solving on linear equations:**

**Example 1:**

(1x + 2) / (x + 3) = 1

(1x + 2) =1 (2x+ 3)

⇒1x + 2 = 2x + 3

⇒X – 2x = 3 – 2**(Shifting X to L.H.S and constants to R.H.S)**

**We get**

⇒-1x = 1 **(Dividing both side by 3 we get)**

**⇒Answer ****X= -1.**

** **

**Example 2:**

3x – 10 = 6x – 19

3x – 10 = 6x – 19

⇒ 3x – 6x = – 19 + 10 **(shifting 6x to L.H.S and -10 to R.H.S)**

⇒ 3x = 9

⇒ 3x/3 = 9/3 **(Dividing both sides by 3)**

Answer x = 3

**Example 3:**

**Solve the value of x**

5 (2x – 9) – 4x= 5 – 2x

⇒10x –45 – 4x = 5 – 2x

⇒10x -4x -2x =5 -45 **(Shift all variables to L.H.S and all constants to R.H.S)**

⇒4x = 40 **(Dividing both side by 10)**

Answer ⇒X= 10

**Linear equation different form**

There are many forms of linear equation but usually the equations have constant like 1, 2, 3…. & have simple variables like x-axis and y-axis.

**Examples of non Linear equations**

Linear functions have a constant slop while plotting on graphs it gives us straight line. While nonlinear functions have slop that varies between points.

⇒X^{2} – 4 = 0

⇒X^{2} /2 = 10

**Linear Equation General Form**

**Ax + By + C = 0**

**Note:**A and B can’t be zero at same time.

General form is not always useful.

**Example**

⇒4x + 2y + 4 = 0

Here

A = 4

B = 2

C = 4

**Slop-intercept form**

The equation slop intercept form is

**Y=mx+b**

Where**“b”** is the y-intercept and **“m”** is the slop of the points. Slop is always same if we change the order of operations.

**Example 1**

Y = -13 + 7x.

**Step1**

**Y = -13 + 7x.**

⇒**For solving y value we Put x=0 we get**

Y = – 13 + 7(0).

Y = – 13.

**Step 2**

**For solving x we put y=0**

⇒0= – 13 + 7x.

⇒13 = 7x**(Dividing both side by 7)**

X= 13/7

The slop interpret form of equation is (0, -13) and (13/7, 0).

**Point slop form**

The equation of a straight line of “**Point-slop**” is

**Y – Y1 =m(X – X1)**

Where X1 and Y 1 are given values and m is the slop of the line.

(X, y) are the unknown values.

**Equations of point slop form**

Where y1 and x1 is given m is slop of the equations and x and y is unknown values.

- Y – 2 = 2(x-3)
- y + 4= ½(x − 8)
- y + 4= 4(x + 10)