Last updated on July 14th, 2024 at 05:32 am

** The Equation Of The Line **

In geometry, the equation of the straight line is widely used to perform various tasks. It is an algebraic form to represent the coordinates points of the line. It is helpful for determining whether the points are lies on the line or not.

The linear equation of the line can be written through different techniques like slope intercept form, point slope form, and x & y intercept form. In this article, we’ll explore the definition of the equation of straight line and how to determine it with the help of point slope form.

**What is the equation of the line?**

The equation of the line is the most common form of the line that is frequently used in geometry. It is also known as the equation of the straight line and linear equation of the line. it can be formed with the help of x & y coordinates points of the line and the slope of the line.

**Slope of the line **

The slope of the line is an essential part of the equation of the line. Let us discuss the slope of the line. The slope of the line is the measure the steepness of the line. it is helpful in describing the sharpness and the direction of the line.

It is the rise over run of the given coordinate points of the line. The rise is referred to the change in the values of y coordinate while the run is referred to the change in the values of c coordinate. It is denoted by “m”.

Slope = m = rise/run

Slope = m = change in y / change in x

**Slope = m = (y**_{2}** – y**_{1}**) / (x**_{2}** – x**_{1}**)**

The slope of the line will remain unchanged throughout the line once it determined. It can be expressed in fractions, numerical values, or tangent of the angle. Now we’ll discuss the point slope form.

**Point slope form**

In algebra, the linear equation of the line needs the slope of the line “m” and a point on the line to express itself in the form of the point slope form. The required point of the line will be (x_{1}, y_{1}) and the slope of the line will be the rise over run of the given points of the line.

The point of the line that is required for the equation of the line must be a numerical values and the slope of the line is the rise/run and can be positive, negative, zero, or undefined. The general expression of the point slope form that is helpful in expressing the linear equation of the line.

**(y – y**_{1}**) = m(x – x**_{1}**)**

- “m” is the rise/run of the line.
- (x
_{1}, y_{1}) the point if the line that is required for deterring the equation of the line. - (x, y) the fixed point of the line.

The expression of the point slope form is linear and can be considered an equation of the line. But it is general form, the point and the slope of the line must be evaluated in it to determine the accurate equation of the line.

**How to determine line equation through point slope form?**

The equation of the line will be evaluated easily with the help of general expression of the point slope form by substituting the point and the slope of the line. Let’s take a few examples to learn how to determine the line equation.

**For two points**

**Example 1**

Determine the linear equation of the line through point slope form if the points of x & y coordinate are: (4, 8) & (2, 3)

**Solution**

**Step 1:** First of all, take the given coordinate points of x & y.

x_{1} = 4, x_{2} = 2, y_{1} = 8, y_{2} = 3

**Step 2:** Now determine the rise/run (m) of the line with the help of x & y coordinate points of the line.

Slope of the line = m = rise/rum = (y_{2} – y_{1})_{ }/ (x_{2} – x_{1})

Slope = m = (3 – 8) / (2 – 4)

Slope = m = (-5) / (-2)

Slope = m = 5 / 2

Slope = m = 2.5

**Step 3:** Now write the expression of the point slope form.

y – y_{1} = m (x – x_{1})

**Step 4:** put the coordinate points of the line and calculated slope to the above expression of the point slope form to get the straight line equation.

y – y_{1} = m (x – x_{1})

y – (8) = 2.5 * (x – 4)

y – 8 = 2.5 * x – 2.5 * 4

y – 8 = 2.5x – 10

y – 8 – 2.5x + 10 = 0

y – 2.5x + 2 = 0

2.5x – y – 2 = 0

**Example 2**

Determine the linear equation of the line through point slope form if the points of x & y coordinate are: (-1, 8) & (21, -13)

**Solution**

**Step 1:** First of all, take the given coordinate points of x & y.

x_{1} = -1, x_{2} = 21, y_{1} = 8, y_{2} = -13

**Step 2:** Now determine the rise/run (m) of the line with the help of x & y coordinate points of the line.

Slope of the line = m = rise/rum = (y_{2} – y_{1})_{ }/ (x_{2} – x_{1})

Slope = m = (-13 – 8) / (21 – (-1))

Slope = m = (-13 – 8) / (21 + 1)

Slope = m = (-21) / (22)

Slope = m = -21/22

**Step 3:** Now write the expression of the point slope form.

y – y_{1} = m (x – x_{1})

**Step 4:** put the coordinate points of the line and calculated slope to the above expression of the point slope form to get the straight line equation.

y – y_{1} = m (x – x_{1})

y – (8) = -21/22 * (x – (-1))

22(y – 8) = -21 * (x + 1)

22y – 176 = -21x – 21

22y – 176 + 21x + 21 = 0

22y + 21x – 155 = 0

The above problems can also be evaluated with the help of online point slope form calculator to avoid time-consuming calculations.

The above problem solved by [https://www.allmath.com/pointslopeform.php]

**For 1 point and slope **

**Example **

Determine the linear equation of the line through point slope form if the point of the line is (15, 20) and slope of the line is -10.

**Solution**

**Step 1:** First of all, take the given coordinate point of the line and slope of the line.

Slope = m = -10

x_{1} = 15

y_{1} = 20

**Step 2:** Now write the expression of the point slope form.

y – y_{1} = m (x – x_{1})

**Step 3:** put the coordinate points of the line and slope to the above expression of the point slope form to get the straight line equation.

y – y_{1} = m (x – x_{1})

y – 20 = -10 * (x – 15)

y – 20 = -10 * x – (-10) * 15

y – 20 = -10 * x + 10 * 15

y – 20 = -10x + 150

y – 20 + 10x – 150 = 0

y + 10x – 170 = 0

10x + y – 170 = 0

**Summary**

Now you can grab all the basics of the equation of the line and how to determine it through point slope form. We have discussed the equation of the line, slope, point slope form, and examples of the point slope form.