The slope of a Line

Introduction: 

Mathematics is considered to be one of the many crucial subjects for anybody preparing for IIT-JEE. Both IIT-JEE Mains and Advanced involve testing of maths in a straightforward way. Elementary mathematics is the basis for understanding physics too. Hence, it is essential to understand the mathematical concepts in a crystal-clear manner. A vital topic in CBSE Class 11 Maths is the slope of a line. This topic is asked in the exam in several ways and has a lot of weightage, directly or indirectly. Whenever it comes to planes, parallel lines, perpendicular lines, angles, etc., the idea of slope gets highlighted somewhere or the other. This article brings to you the basic definition, significance, formulas, and examples relating to the slope of a line. 

What is meant by the slope of a line?

A slope generally denotes the steepness or slant of a surface. In mathematical terms, the slope of a given line is defined by the fraction of the shift in the y-axis to the shift in the x-axis. Generally, the slope (m) of a line provides the magnitude of its steepness along with its direction. The slope of a given straight line between two points (x1,y1) and (x2,y2) can be easily calculated by finding the difference between the coordinates of the points. The letter’ m’ usually represents the slope. Hence, the change in y-coordinate concerning the shift in x-coordinate is given by,

m = shift in y/shift in x = Δy/Δx (where ‘m’ represents the slope of the line).

In another way, the slope of the line can also be calculated as

tan θ = Δy/Δx

A positive and negative slope.

If the slope of a line is positive, it shows that the line directs up as we move along. On the other hand, if the Slope value is negative, the line goes down as we move along the x-axis. 

1. Positive Slope

y = mx + b (m > 0)

y increases as x increases

  2. Negative Slope

y = mx + b (m < 0)

y decreases as x increases

For example, line 6=4x+2 has a positive slope where m=4. On the contrary, the line 45=15-3x has a negative slope where m=-3. 

Slope of a line formulas

Graphical interpretation of the slope of a line

If a graph is provided in the question and the slope to be calculated, it can be done effortlessly. All the slope formulas have been derived and formulated using graphs. The graphical representation of the problem makes it easier to understand and find a feasible solution. For example, if the diagram depicts a line intersecting both the x-axis and y-axis in the first quadrant, the slope can be found out using these steps:-

1. Mark both the points on the graph.
2. Write the coordinates at the place where the line intersects the x and y-axis. For example, if the line crosses the y-axis at 4, the coordinates are (0,4). Similarly, if the line intersects the x-axis at 6, then the coordinates of that point are (6,0).
3. Now, we have the values of y2,y1,x2 , and x1.
4. Using the two-point slope formula, the slope can be easily calculated. The formula is:

              m = tan θ = (y2 – y1) / (x2 –  x1)

          For the above example, the slope will be ‘m’= 0-4/6-0=-2/3. 

Solved examples

Let us have a look at some solved examples:

Problem 1:

Find the line slope between the two given points, P(–2, 4) and Q(0, –1).

Solution:

Given, P(–2, 4) and Q(0, –1) are the two points.

Therefore, slope of the line, 

m = (-1-4)/0-(-2) = -5/2 = -2.5

Problem 2:

The slope of a line is 3/5. One point is (9,-9), and the other is (2x,-3). Find x.  

Solution:

Let (9, -9) be point 1, and let (2x, -3) be point 2. It is given already that m=3⁄5. 

Therefore, we have

3⁄5=(-9+3)⁄(9-2x)

3⁄5=-6⁄(9-2x)).

Cross-multiplying, 

3(9-2x)=-30

x=19/2

Problem 3:

Find the slopes m1 ​and m2 of the straight lines y=3x-3 and x+3y+5=0. Calculate =m1 . m2

Solution:

For line 1,

y=3x-3

Comparing it with y=mx+c, we get m as 3.

For line 2,

x+3y+5=0

3y=-x-5

y=-x/3 -5/3

Comparing it with y=mx+c, we get m as -1/3.

Hence, m1.m2= 3.(-1/3)=-1.

Tips to prepare CBSE Class 11: Slope of a line in a better way:

• Make a timetable before beginning with the topic. The schedule should be prepared to keep in mind the vast IIT-JEE syllabus, and time should be allotted to every subject.
• It is always better to write what you understand, definitions, concepts, theorems, or formulas. The best way to memorise any fact or detail is by making physical notes of that topic. Self-written formulas and examples are easy to understand and recall.
• Give equal time to all topics. As ‘slope of a line’ is related to many other topics and may require additional practice, keep a revision slot.
• Do not forget to practice many quizzes, mock tests, and previous years’ papers. This gives an idea of the weightage and essential topics from CBSE Class 11: Slope of a line.
• Practising maths can be monotonous sometimes. Hence, frequent breaks can help keep you focused. 

Conclusion:

This article summarises the entire CBSE Class 11 chapter: Slope of a line, with sufficient examples. By now, it might have become clear that this concept holds a lot of importance while preparing for IIT-JEE Mains and Advanced. Many textbooks, reference books, and e-resources are available to help you prepare better.