Graphing and Linear Equations Formulas | List of Graphing and Linear Equations Formulas You Should Know - BYJUS

# Graphing and Linear Equations Formulas

When we represent a linear equation in a coordinate plane, then the representation is known as graphing a linear equation. The graph of a linear equation is a straight line. We graph equations by plotting the ordered pairs that satisfy the equation on the coordinate plane and passing a straight line through them....Read MoreRead Less

### Formula to find the Slope of a Line

When we represent a linear equation in a coordinate plane, then the representation is known as graphing a linear equation. The graph of a linear equation is a straight line. We graph equations by plotting the ordered pairs that satisfy the equation on the coordinate plane and passing a straight line through them.

The slope of a line is the ratio of rise in the y-axis to the run in the x-axis. In simple words, slope is the ratio of the change in y coordinates to the change in x coordinates. In general, slope is denoted by ‘m’.

Consider two points $$(x_1,~y_1)$$ and $$(x_2,~y_2)$$ that satisfies the equation $$y~=~mx+b$$. The slope of the line will be given by :

$$m=~\frac{\text{rise}}{\text{run}}$$

$$=~\frac{\text{change in y}}{\text{change in x}}$$

$$=~\frac{y_2~-~y_1}{x_2~-~x_1}$$

### Slope of Vertical Line

A vertical line is parallel to the y-axis. That is, there will be no change in the x-axis. The slope will then be:

m = $$~\frac{\text{rise}}{\text{run}}$$

= $$~\frac{\text{change in y}}{\text{change in x}}$$

= $$~\frac{y_2~-~y_1}{x_2~-~x_1}$$

= $$~\frac{y_2~-~y_1}{0}$$

We know that division by zero is undefined. Hence the slope of a vertical line can not be defined.

### Slope of Horizontal Line

A horizontal line is parallel to the x-axis. So, there will be no change in the y-axis. The slope will be:

m = $$~\frac{\text{rise}}{\text{run}}$$

= $$~\frac{\text{change in y}}{\text{change in x}}$$

= $$~\frac{y_2~-~y_1}{x_2~-~x_1}$$

= 0

The slope of a horizontal line is 0.

### Rapid Call

Slope of line:

m = $$~\frac{\text{rise}}{\text{run}}$$

= $$~\frac{\text{change in y}}{\text{change in x}}$$

= $$~\frac{y_2~-~y_1}{x_2~-~x_1}$$

• The slope of a vertical line is undefined
• The slope of a horizontal line is 0.

### Solved Examples

Example 1: Calculate the slope of the given line.

Solution :

Let $$(x_1,~y_1)=(-3,~4)$$ and $$(x_2,~y_2)=(3,~-5)$$.

m = $$~\frac{y_2~-~y_1}{x_2~-~x_1}$$

= $$~\frac{-~5~-~4}{3-~(-~3)}$$

= $$~\frac{-~9}{6}$$

= $$~\frac{-~3}{2}$$

Example 2: Identify the slope of the given line.

Solution :

This is a horizontal line hence the slope is 0.

Applying the slope formula,

m = $$~\frac{y_2~-~y_1}{x_2~-~x_1}$$

= $$~\frac{3~-~3}{4~-~(-~2)}$$

= 0

Therefore, the slope is 0.

Example 3: Calculate the slope of the given line.

Solution :

The slope of a vertical line is undefined.

Applying the slope formula,

m = $$~\frac{y_2~-~y_1}{x_2~-~x_1}$$

= $$~\frac{3~-~0}{5~-~5}$$

= $$~\frac{3}{0}$$

Division by zero is undefined. So, the slope is undefined.

Example 4: The table shows the distance y(in km) traveled by car in x minutes. The points in the table form a straight line. Find the slope of the line.

x

3

6

9

12

y

5

7

9

11

Solution :

First, each of the ordered pairs is plotted on the coordinate plane. These points are joined by a straight line.

To find the slope, take any two points from the table and apply the slope formula.

Use points $$(x_1,~y_1)=(3,~5)$$ and $$(x_2,~y_2)=(6,~7)$$

m = $$~\frac{y_2~-~y_1}{x_2~-~x_1}$$

= $$~\frac{7~-~5}{6~-~3}$$

= $$~\frac{2}{3}$$

The slope is $$\frac{2}{3}$$, which means that the distance traveled by car increases by 2 kilometers every 3 minutes, or the car travels 23 kilometers every minute.

The standard form of line is Ax + By = C.

If the slope of line $$l_1$$ and $$l_2$$  is $$m_1$$ and $$m_2$$ respectively and line $$l_1\left | \right |l_2$$. Then the value of $$m_1$$ and $$m_2$$ is same i.e., $$m_1~=~m_2$$.

If the slope of line $$l_1$$ and $$l_2$$ is $$m_1$$ and $$m_2$$ respectively and line $$l_1~\perp~ l_2$$. Then $$m_1\cdot~m_2=-1$$.

The point slope form of a straight line is given by :

$$y~-~y_1~=~m(x~-~x_1)$$.

The graph of a linear equation is a straight line.