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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

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 :

\(m=~\frac{\text{rise}}{\text{run}}\)

\(=~\frac{\text{change in y}}{\text{change in x}}\)

\(=~\frac{y_2~-~y_1}{x_2~-~x_1}\)

** **

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.

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.

**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.

**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.

Frequently Asked Questions

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.