Lesson 5 Equation of a Line Linear Equation

Lesson 5 Equation of a Line

Linear Equation � Representation � Let us get in touch with lines and their representation. Every linear equation represents a unique line, for example 3 x + 4 = 7 represents a line. It is a linear equation with one variable. Another equation of the form 2 x + 3 y = 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.

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.

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.

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 3 y = 2 x, 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. �

Slope of a Line

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.

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

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 = m 1 x + c and y = m 2 x + c, then they are parallel if m 1 = m 2.

Parallel Lines � Example: y = 2 x + 3, y = 2 x + 7 are parallel as we observe that their slopes are equal (slope is 2 here) Let us take another example, 3 x + 4 y = 8 and 6 x + 8 y = 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 3 x + 4 y = 8 is -3/4 and � Slope of the line 6 x + 8 y = 17 is = -3/4 � So, the lines are parallel as their slopes are equal.

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 = m 1 x + c and y = m 2 x + c, then the lines are perpendicular if m 1 * m 2 = -1. That is if two lines are perpendicular, then the product of their slopes is -1.

Perpendicular Lines � Example: Let us consider the lines y = 3 x + 4 and y = x/3+7 Slope of 1 st line = m 1 = 3 Slope of 2 nd line = m 2 = -1/3 m 1 * m 2 = 3 * = -1 � As the product of the slopes of the given lines is -1, they are perpendicular

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 = -5 x + 4.

Equation of a Line

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 = -7 x represents a line passing through the origin

References � Online Free SAT Study Guide: SAT Guide � http: //www. proprofs. com/sat/study- guide/index. shtml