What is the equation of line and how to calculate it?

In mathematics, the equation of a line is a way of representing a line using an algebraic equation. There are three main forms of the equation of a line: slope-intercept form, point-slope form, and x & y-intercept form.

In this blog post, we will explain the first form of the equation of a line. We will discuss the definition, methods, and examples of slope intercept form, point-slope form, and x & y-intercept form along with examples.

What is the equation of a line?

In a coordinate system, the equation of line represents the set of points forming the line. In an algebraic equation, the numerous points that together form a line in the coordinate axis are represented by the variables x and y.

There are various ways to calculate the equation of a line. Such as slope intercept form, point slope form, and x & y intercept form. The slope intercept form of the equation of a line is y = mx + b. This forms the basis for solving equations of lines in one dimension.

To find the equation of a line given a point and a slope, use the point slope form of the equation which is y – y₁ = m(x – x₁).

To find the equation of a line given two points, use the x & y intercept form which is y = mx + b. Knowing these forms can be helpful in many situations, such as when you are graphing linear data or when you are trying to solve an equation.

Forms of the equation of line

There are a number of different forms that the equation of a line can take. Such as

1. Slope intercept form
2. Point slope form
3. X & Y intercept form

1. Slope intercept form

The first form of the equation of a line is called the slope-intercept form. This is the simplest form, and it looks like this y = mx + b. In this form, y represents the height (or magnitude) of the curve, x represents the location on the curve, and m represents the slope of the curve. The value for b depends on where x is located on the curve.

The slope intercept form is especially useful when graphing linear data. When you are plotting points on a graph, it can be helpful to know the equation of the line that connects these points. To find this equation, use the point slope form of the equation.

This form can be found by subtracting each point’s y-value from its x-value, then dividing that result by the y-1 value of that point.

Another situation where knowing the slope intercept form can be helpful is when you are trying to solve an equation. To do this, use the point slope form of the equation and solve for mx. Once you have solved for mx, you can use it to determine y or plot it on your graph.

Example: For slope intercept form

Find the equation of straight line with the help of slope intercept form if the given coordinate points of the line are:

(x₁, y₁) = (2, -8) & (x₂, y₂) = (18, 24)

Solution

Step 1: First of all, write the given coordinate points of the line.

x₁ = 2, x₂ = 18, y₁ = -8, y₂ = 24

Step 2: Now evaluate the slope of the line with the help of coordinate points of x & y.

Slope = m = [y₂ – y₁] / [x₂ – x₁]

Substitute the given values

Slope = m = [24 – (-8)] / [18 – 2]

Slope = m = [24 + 8] / [18 – 2]

Slope = m = [32] / [16]

m = 32/16

m = 2

Step 3: Now take the formula of the slope intercept form and evaluate the y-intercept form of the line by placing the coordinate points and slope of the line.

The general expression of the slope intercept form.

y = m * x + b

Substitute m = 2 and (x₁, y₁) = (2, -8)

-8 = 2 * (2) + b

-8 = 4 + b

-8 – 4 = b

b = -12

Step 4: Now to find the equation of the line, place slope and y-intercept values to the general expression of the slope intercept form.

y = m * x + b

y = 2 * x + (-12)

y = 2x – 12

Hence, the above expression is the linear equation of the line.

The slope intercept form calculator can be used to find the equation of a line if the coordinates of two points on the line are given.

2. Point slope form

The second form of the equation of a line is called the coordinate form. This form looks like this (x1, y1), (x2, y2) … (xn, yn). In this form, xi represents the coordinates of point X on a curve and yj represents the coordinates of point Y on a curve.

The equation of a line is most commonly written in slope intercept form. This form is used when you know the slope and coordinates of one point on the line. For example, if you are given the slope and y-intercepts for a line that passes through Point A and Point B, then you can use the point slope form to find the equation of that line.

Point slope form is also used when you don’t know the coordinates of one point on the line. In this case, you can use the x-intercept form to find where the line crosses both the x-axis and y-axis.

Example 2: For point slope form

 Calculate the linear equation of the line by using the slope intercept form (two point’s method). If the coordinate points of the line are:

(x₁, y₁) = (-2, -26) & (x₂, y₂) = (12, 14)

Solution

Step 1: Take the given information of the given coordinate points of the line.

x₁ = -2, x₂ = 12, y₁ = -26, y₂ = 14

Step 2: First of all, evaluate the slope (steepness) of the line with the help of the coordinate points of the line. You can also find the slope of a line through a slope calculator.

Formula of calculating the slope of the line  

Slope = m = [y₂ – y₁] / [x₂ – x₁]

Substitute the given values

Slope = m = [14 – (-26)] / [12 – (-2)]

Slope = m = [14 + 26] / [12 + 2]

Slope = m = [40] / [14]

m = 40/14

m = 20/7 = 2.86

Step 3: Now write the general expression of the point slope form for calculating the equation of the line.

(y – y₁) = m * (x – x₁)

Step 4: Now put the calculated value of the slope of the line to the general expression of the point slope form and any pair of the coordinate’s points of the line to determine the straight line equation of the line.

The formula of the point slope form.

(y – y₁) = m * (x – x₁)

Substitute m = 2.86 and (x₁, y₁) = (-2, -26)

(y – (-26)) = 2.86 * (x – (-2))

(y + 26) = 2.86 * (x + 2)

(y + 26) = 2.86 * x + 2.86 * 2

y + 26 = 2.86x + 5.72

y + 26 – 2.86x – 5.72 = 0

y – 2.86x + 20.28 = 0

2.86x – y – 20.28 = 0

3. X & Y intercept form

The x & y intercept form of the equation of a line is useful when finding the equation of a line given the coordinates of two points on the line. This form can be used in conjunction with other forms, such as the slope intercept form and point slope form, to calculate more precise equations.

Additionally, this form is useful when graphing linear equations. By knowing which form to use for each set of coordinates, it becomes easier to understand and interpret graphs relating to linear equations.

The point slope form is another way to find the equation of a line given the coordinates of two points on the line. The point slope form takes into account both points’ slopes in order to calculate an equation for that line.

Conclusion

There are many different forms that the equation of a line can take, each with its own benefits and drawbacks. In this blog post, we’ve looked at three of the most common forms: slope-intercept form, point-slope form, and x & y intercept form.