Counter Examples
Mathematics Dictionary
Dr. K. G. Shih
Linear Functions and Equations
Subjects

  • Read Symbol defintion
  • Q01 | - Highlight of this lesson
  • Q02 | - What is linear function ?
  • Q03 | - What is linear equation ?
  • Q04 | - Vertical line and horizontal line
  • Q05 | - Study the line y = 2*x + 1
  • Q06 | - What is the inverse of line y = m*x + n ?
  • Q07 | - Find the inverse of line y = 2*x + 3 ?
  • Q08 | - Find intersection of line y = 2*x + 3 with its inverse ?
  • Q09 | - Find intersection of line y = 2 with its inverse ?
  • Q10 | - Solve abs(x + 2) = 3 where abs means absolute value
  • Q11 | - Graph of y = |x-1| + |x+1| = c
  • Q12 | - Solve |x-1| + |x+1| = 3
  • Q13 | - Angle between two lines
  • Q14 | - Two lines are parallel
  • Q15 | - Angle between y = 2*x + 3 and its inverse
  • Q16 | - Intrsection between y = 2*x + 3 and its inverse
Answers


Q01. Hightlight of this lesson

Study topics
    * Graphic proof of inverse of y = m*x + n is x = m*y + n (Q6)
    * How to solve abs(m*x + n) = p (See Q09)
    * Graphic solution of (m*x + n) > p
    * Graphic solution of (m*x + n) < p
How to use program ABH ?
  • Program ABH - Graphic Calculator Examples
      * No download and run at current location
      * Click 01 Linear Function and Equation in upper box
      * Click program number in lower box
      * Now it is in graphic mode
      * Return to menu mode - Click Back command
  • Visit the 130 graphic samples in Program ABI
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Q02. What is linear function ?

Definition
    * Linear function : Y = m*x + n
    • Where m is the slope of the line and n is y-intercept
    • The line makes angle A with x-axis and then tan(A) = m
    • If we know slope m then we can find A by arcTan(m)
    • If we know slope m then we can also find A by construction
    * Linear function is a straight line
    * Linear function has an inverse function
    * Linear function is 1st degree polynomial
Graphic properties
  • y = n is the y-intercept when x = 0
  • x = -n/m is the zero value
  • Slope is m
  • If m GT 0, the line is increasing
  • If m LT 0, the line is decreasing or y = n
  • If m EQ 0, the line is parallel to x-axis
  • If m EQ infinite, the line is parallel to y-axis or x = b
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Q03. What is Linear Equation ?

Definition
    * Linear equation : m*x + n = 0
    * The solution of linear equation is called root
    * The root is r = -n/m and m not equal to 0
    * The root is same the zero of Y = m*x + n
    * Liear equation has only one real root
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Q04. Study the line y = 2

Study the line x = 2 ?
    * It is a vertical line which cuts the x-axis at x = 2
    * It is parallel to y-axis and zero value is 2
    * Its slope is infinite and no y-itercept
    * It makes angle with x-axis A = 90 degree
    * Graphic : Program ABH Topic = 01 and program 02
    * Input : m = 0 and n = 2 for x = m*y + n
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Q05. Study the line y = 2*x + 1

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Q06. What is the inverse of line y = m*x + n ?
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Q07. Find the inverse of line y = 2*x + 3 ?
    * Graph of y = 2*x + 3 :
      Program ABH Topic = 01 and program 01
      Input : m = 2 and n = 3 for y = m*x + n

    * Graph of x = 2*y + 3 :
      Program ABH, Topic = 01 and program 02
      Input : m = 2 and n = 3 for x = m*y + n

    * Graph of x = (y - 3)/2 :
      Program ABH, Topic = 01 and program 02
      Input : m = 0.5 and n = -1.5 for x = m*y + n

    * Graph of y = (x - 3)/2 :
      Program ABH, Topic = 01 and program 01
      Input : m = 0.5 and n = -1.5 for x = m*y + n

    * From above examples we see that the inverse is
      The inverse is y = (x - 3)/2 or
      The inverse is x = 2*y + 3
      Hence the inverse of y = m*x + n is x = m*y + n
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Q08. Find intersection of line y = 2*x + 3 with its inverse ?
    * Program ABH Topic = 01 and program 03
    * Input : m = 2 and n = 3 for y = m*x + n
    * Angle of y = 2*x + 3 making with x-axis is A1 = arctan(2)
    * Angle of inverse making with x-axis is A2 = arctan(1/2)
    * Angle between line and inverse is A1 - A2

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Q09. Find intersection of line y = 2 with its inverse ?

    * Program ABH Topic = 01 and program 03
    * Input : m = 0 and n = 2 for y = m*x + n
    * Angle of y = 2 making with x-axis is A1 = 90
    * Angle of inverse making with x-axis is A2 = 0
    * Angle between line and inverse is A1 - A2 = 90
    * What is inverse of y = 2 ? Answer : x = 2

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Q10. Solve abs(x + 2) = 3 where abs means absolute value

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Q11. Graph of y = abs(x - 1) + abs(x + 1)

  • Graphic is in Program ABH 07 01
  • This equations contains 3 equations :
    • When x GT 1, y = 2*x.
    • When x between -1 and 1, y = 2.
    • When x LT -1, y = -2*x.
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Q12. Solve abs(x - 1) + abs(x + 1) = c

  • Solutions are in Program ABH 07 02
  • Example Solve |x-1| + |x+1| = 3.
    • This equations contains 3 equations :
      • When x GT +1, |x-1| + |x+1| GT 2.
      • When x between -1 and 1, |x-1| + |x+1| = 2.
      • When x LT -1, |x-1| + |x+1| GT 2.
    • Hence there are two roots.
      • When x GT +1, y = 2*x. Hence +2*x = 3 and x = +1.5.
      • When x LT -1, y =-2*x. Hence -2*x = 3 and x = -1.5.
  • Example Solve |x-1| + |x+1| = 1.
    • This equations contains 3 equations :
      • When x GT +1, |x-1| + |x+1| GT 2.
      • When x between -1 and 1, |x-1| + |x+1| = 2.
      • When x LT -1, |x-1| + |x+1| GT 2.
    • Hence there are no solutions.
  • Example Solve |x-1| + |x+1| = 2.
    • This equations contains 3 equations :
      • When x GT +1, |x-1| + |x+1| GT 2.
      • When x between -1 and 1, |x-1| + |x+1| = 2.
      • When x LT -1, |x-1| + |x+1| GT 2.
    • Hence the solution is x between -1 and 1.
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Q13. Angle between 2 lines

  • Construction
    • Draw line AB and line CD which meets at E.
    • Draw axis Ox and Oy.
  • Definition :
    • Slope of line = tan(U)
    • where U is angle of line making with x-axis.
  • Slope of line AB is m1 = tan(A1). A1 is angle of AB making wiht x-axis.
  • Slope of line CD is m2 = tan(A2). A1 is angle of AB making wiht x-axis.
  • Hence angle between ab and CD is AEC = A1 - A2
  • Since tan(AEC) = tan(A1-A2) = (tan(A1)-tan(A2))/(1+tan(A1)*tan(A2))
  • Hence tan(AEC) = (m1 - m2)/(1 + m1*m2)
  • If line AB is perpendicular to CD, then
    • Angle AEC = 90 degrees.
    • tan(90) = infinites.
    • Hence 1 + m1*m2 = 0. This the condition for two line perpemding.
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Q14. Two lines are parallel

Construct a line parallel to other line
  • Construction
    • Draw line AB
    • Draw a line EF which cut AB at P
    • Draw a point Q on line EF
    • Draw angle PQC = 180 - angle QPB
    • QC is the line parallel to AB
  • Condtions for parallel
    • Alternative angles are equal (Angle APQ = Angle PQD).
    • Coresponding angles are equal (Angle EPB = Angle PQD).
    • Angle PQD + angle QPB = pi.
Distance between two parallel lines
  • line 1 is y = a*x + b
  • Line 2 is y = a*x + d
  • Distance between two lines is (b - d)*cos(A)
  • Where A is angle of line making with x-axis and A = arctan(a)
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Q15. Angle between y = 2*x + 3 and its inverse

Angle of line y = 2*x + 3 making with x-axis
  • The angle is A1 = arctan(m) where m is the slope and m = 2
  • Hence A1 = 63.435 degrees
Angle of inverse of line y = 2*x + 3 making with x-axis
  • The inverse is x = 2*y + 3 or y = x/2 - 3/2
  • The angle is A2 = arctan(m) where m is the slope and m = 1/2
  • Hence A2 = 26.565 degrees
Angle between y = 2*x + 3 and its inverse
  • A = A1 - A2 = 63.435 - 26.565 = 36.87 degrees
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Q16. Intrsection between y = 2*x + 3 and its inverse

Answer
  • Find intersection of y = 2*x + 3 and x = 2*y + 3
  • Hence y = 2*(2*y + 3) + 3
  • Hence -3*y = 9 and y = -3
  • Hence x = -3
  • Intersection point is at (-3, -3)
Go to Begin
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