1. Lesson 5
Equation of a Line

2. Linear Equation Representation
Let us get in touch with lines and their representation. Every linear equation represents a unique line, for example 3x + 4 = 7 represents a line. It is a linear equation with one variable. Another equation of the form 2x + 3y = 7 also represents a line. Here this is a linear equation with two variables.
In general we can say an equation of the form ax + by + c = 0 where a and b are not simultaneously zero represents a straight line in a plane.

3. Inclination of a Line Explanation
To understand the concept of slopes of a line, let us first know about inclination or steepness of a line. Let us take the example of a ladder placed against a wall.
If a person has to climb the ladder it must be kept at an angle with the ground.
This angle is known as inclination or steepness at which the ladder is placed against the wall.
We often hear the work that the steps of a staircase are too steep.

4. Inclination of a Line
All these can be related to lines also. The angle made by a line with x-axis in the anti clock wise direction is called its steepness or inclination. The lines l and m given below are making angles and respectively with x-axis.
These figures and examples give a better idea of the slope, hence it is included, can be deleted if not required.

5. Slope of a Line Illustration
Slope is a concept that is related to steepness of a line. If we look at the graph of the line 3y = 2x, we will observe that this line passes through the origin.
If we take any point except the origin on this line, the ratio between its y-coordinate and x-coordinate is constant.

6. Slope of a Line

7. Slope of a Line
(Ratio of y and x coordinates of points A, B and C.) This constant is called the slope of the line.
Therefore, y = mx represents a line passing through the origin whose slope is m.

8. Slope of a Line Example: Find the slope of the line 5y = -7x Solution:
5y = -7x or y = -7x/5  Slope (m) = -7/5

9. Parallel Lines Condition
If two lines in a plane are given, we can find out if they are parallel or not using certain conditions. If two lines in a plane are parallel then their slopes are equal.
If we have two lines of the form, y = m1x + c and y = m2x + c, then they are parallel if m1 = m2.

10. Parallel Lines
Example: y = 2x + 3, y = 2x + 7 are parallel as we observe that their slopes are equal (slope is 2 here) Let us take another example, 3x + 4y = 8 and 6x + 8y = 17. As soon as we see this we cannot say if these lines are parallel, hence there has to be a method to check for parallelism.
Slope of the line 3x + 4y = 8 is -3/4and
Slope of the line 6x + 8y = 17 is = -3/4
So, the lines are parallel as their slopes are equal.

11. Perpendicular Lines Condition
As we tested for parallelism we can also find out if two given lines are perpendicular or not. If we have two lines of the form, y = m1x + c and y = m2x + c, then the lines are perpendicular if m1 * m2 = -1. That is if two lines are perpendicular, then the product of their slopes is -1.

12. Perpendicular Lines
Example: Let us consider the lines y = 3x + 4 and y = x/ Slope of 1st line = m1 = 3 Slope of 2nd line = m2 = -1/3   m1 * m2 = 3 * = -1
As the product of the slopes of the given lines is -1, they are perpendicular

13. Equation of a Line Slope Intercept Form
Equation of a line can be found in different ways based on the available information. Let us look at them one-by-one.
The slope-intercept form of a line:
If a line has a slope m with y-intercept c then the equation of the line is in the form y = mx + c. If we want to find out the equation of a line whose slope is -5 and c is 4, hence, equation of the line will be, y = -5x + 4.

14. Equation of a Line

15. Equation of a Line
Let us think, how the equation of a line will be if it passes through the origin. If a line passes through the origin then it does not make any intercepts on the axis, therefore making c zero. Hence, the equation of a line passing through the origin will be y = mx + 0 or y = mx. Example: y = -7x represents a line passing through the origin

16. References Online Free SAT Study Guide: SAT Guide
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