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Lesson 5

Lesson 5. Equation of a Line. Linear Equation . Representation

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Lesson 5

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  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 example3x + 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 + cand 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 + 4and y = x/3+7Slope of 1st line = m1 = 3Slope 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 • http://www.proprofs.com/sat/study-guide/index.shtml

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