Slope Intercept Form of a Line

Introduction

The slope-intercept form of a straight line is a way of representing a line. It is typically represented by the particular form of a linear equation, i.e, y=mx+c. It stresses upon the slope along with the y-intercept of a given line. 

The slope-intercept form is a commonly prefered way of representing the equation of a straight line. When the slope of a straight line and its y-intercept is known, its equation can be determined. The slope is defined as the angle at which a straight line is inclined to the x-axis. The y-intercept is basically the y-coordinate point where the line makes intersection with the y-axis. 

Definition of the Slope-Intercept Form of a Straight Line

The slope-intercept form provides the basic and simple form of the equation of a straight line. With the slope of the line and the y-intercept known, the equation of a straight line can be easily formulated. 

The slope-intercept form of a straight line represents a straight line in the form of a linear equation. The formula used to represent the equation is as follows:

y = mx + c

Here, y and x are variables. ‘m’ represents the slope of the line while ‘c’ denotes the y-intercept of the line.

Derivation of the Equation of the Slope-Intercept Form of a Line

For the derivation of the formula of the slope-intercept form of a line, we assume a straight line with the slope ‘m’ that intersects the y-axis at the point (0,c), indicating that its y-intercept is ‘c’. Along with this, we also assume that the point (x,y) is an arbitrary or random point present on the line.

We assume that (x1, y1) = (0, c) and (x2, y2) = (x, y).

Now, according to the slope formula, the slope of a line connecting the two given points (x1, y1) and (x2, y2) is given by m = (y2 – y1)/(x2 – x1)

By this formula, we get the slope to be

m = (y – c) / (x – 0)

⇒ m = (y – c) / (x)

After multiplying both sides with x, we get

mx = y – c

Adding the variable ‘c’ on both sides, we get

y = mx + c

The slope-intercept formula of a straight line is derived hereby.

Since, this basic equation of a straight line includes the factors like slope (m) and y-intercept (c), therefore, it is called the slope-intercept form of a straight line. 

Application of the Slope Intercept Form Of A Straight Line 

The slope-intercept formula can be used to find the slope and y-intercept of a line simultaneously. Similarly, when the slope (m) and y-intercept (c) of a straight line are known, then the equation of a straight line can easily be formed. 

Problem: If the equation of a line is given by 5x + 7y + 9 = 0, then find out the slope and y-intercept of the line by applying the slope-intercept formula.

Solution: On rearranging the given equation in the form of the equation y=mx+c, we get:

7y = -5x – 9

Solving the equation:

7y = (-5)x -9

7y = (-5)x + (-9)

⇒ y = (-5/7)x + (-9/7) [taking 7 in to the right hand side of the equation]

Thus we get, the slope, m = -5/7 and y-intercept, c = -9/7.

Examples Of The Slope-Intercept Form Of A Straight Line

Below are a few examples of the formula which is representative of the slope-intercept form-

Example 1: Given the information about a line whose slope is -3 and the y-intercept is 1, we can form its equation as y = -3x + 1.

Explanation: Replacing the values in the equation y=mx+c, we get y = -3x + 1 because here, ‘m’ is -3 and ‘c’ is 1.

Example 2: Another interesting example shows the equation of a line that passes through the origin. 

Explanation: We already know that the coordinate for origin is (0,0). Given that the slope is 3 and the line passes through the origin, we get the y-intercept to be 0.

Putting the values in the equation y = mx + c, given that m = 3 and c = 0, we get

y = 3x + 0

Hence, the line equation is given as y = 3x.

Finding Out The Slope Of A Straight Line

The angle of the slope of a line can be obtained by knowing the value of tanθ. In other words, the angle of inclination can be found out by calculating tanθ. When the coordinates of any two random points are known, then its slope can be determined by a simple formula.

Formula for finding the value of tanθ: If any two random points on a line are given by (x1, y1) and (x2, y2), we can find out the slope by calculating the value of (y2-y1)/(x2-x1).

Converting Standard Form of a Line Equation to the Slope-Intercept Form

Any line equation provided in the standard form can be converted into the slope-intercept form of a straight line by simple reshuffling and balancing. 

The standard form of the equation of a straight line is already known by:

ax + by + c = 0. 

Rearranging the terms for finding the value of ‘y’, we can write the equation as-

by = -ax + (-c)

⇒ y = (-a/b)x + (-c/b)

where (-a/b) denotes the slope of the line and (-c/b) denotes the y-intercept.

Slope Intercept Form of a Straight Line: Important Points To Remember

The slope of a line can be negative. If the line makes an obtuse angle with the positive direction of the x-axis, its slope is negative. Since the value of tan θ becomes negative, the slope also is negative.

A line that passes through the origin will have its equation in the form of y=mx, because in this case, the y-intercept, ‘c’ is zero. So, in the equation, the y-intercept has no value.

Conclusion

The slope-intercept form is helpful to determine the equation of a straight line in the coordinate plane. All the coordinate points the line passes will be valid for the equation. Reversely, any coordinate point that does not fall on the line will not satisfy the equation. The determination of this equation is simple. To determine the slope-intercept form of a given straight line, one needs to know the slope and intercept of the line.