June 18, 2010

Algebra: Root of a Linear Equation In One Unknown (Fractional)



fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots fractional linear equations in one unknown and their roots
Posted by Gerardo Emerzona I

June 3, 2010

Analytic Geometry: Graphs of Intersecting And Parallel Lines



The following illustrated problems are about:

  • intersecting lines (perpendicular and non-perpendicular)
  • parallel lines


All the calculated values are given so as to allow flexibility in presenting the problems in different ways.

1-1 Graphs of Perpendicular Bisector of a Line Segment



Find the point at which the line x - 2y + 3 = 0 bisects perpendicularly the line segment formed by the x-intercept and y-intercept of the line 2x + y - 4 = 0.

graph of a line segment and its perpendicular bisector

  • point of intersection (midpoint of line segment AB): (1,2)
  • slope of the line x - 2y + 3 = 0: m = 1/2
  • angle of inclination of line x - 2y + 3 = 0: 26.6°
  • points on the line x - 2y + 3 = 0: {(-5, -1), (-4, -1/2), (-2, 1/2), (-1, 1), (0, 3/2), (-3, 0), (1, 2), (2, 5/2), (3, 3), (4, 7/2), (5, 4)}
  • slope of the line 2x + y - 4 = 0: m = -2
  • angle of inclination of line 2x + y - 4 = 0: 116.6°
  • points on the line 2x + y - 4 = 0: {(-5, 14), (-4, 12), (-3, 10), (-2, 8), (-1, 6), (0, 4), (2, 0), (3, -2), (4, -4), (5, -6)}


Find the point at which the line x + 5y + 12 = 0 bisects perpendicularly the line segment formed by the x-intercept and y-intercept of the line 5x - y - 5 = 0.

graph of a line segment and its perpendicular bisector

  • point of intersection (midpoint of line segment AB): (1/2, -5/2)
  • slope of the line x + 5y + 12 = 0: m = -1/5
  • angle of inclination of line x + 5y + 12 = 0: 168.7°
  • points on the line x + 5y + 12 = 0: {(-5, -7/5), (-4, -8/5), (-3, -9/5), (-2, -2), (-1, -11/5), (0, -12/5), (-12, 0), (1, -13/5), (2, -14/5), (3, -3), (4, -16/5), (5, -17/5)}
  • slope of the line 5x - y - 5 = 0: m = 5
  • angle of inclination of line 5x - y - 5 = 0: 78.7°
  • points on the line 5x - y - 5 = 0: {(-5, -30), (-4, -25), (-3, -20), (-2, -15), (-1, -10), (0, -5), (1, 0), (2, 5), (3, 10), (4, 15), (5, 20)}


Find the point at which the line 5x + 3y = 0 bisects perpendicularly the line segment AB.

graph of a line segment and its perpendicular bisector

  • point of intersection (midpoint of line segment AB): (-3/2, 5/2)
  • slope of the line 5x + 3y = 0: m = -5/3
  • angle of inclination of line 5x + 3y = 0: 120.96°
  • points on the line 5x + 3y = 0: {(-5, 25/3), (-4, 20/3), (-3, 5), (-2, 10/3), (-1, 5/3), (0, 0), (1, -5/3), (2, -10/3), (3, -5), (4, -20/3), (5, -25/3)}
  • slope of the line segment AB: m = 3/5
  • equation of line segment AB: 3x - 5y + 17 = 0


1-2 Graphs of Perpendicular Lines



Find the point of intersection of lines 8x + 5y + 13 = 0 and 5x - 8y - 3 = 0.

graph of 2 perpendicular lines

  • point of intersection: (-1, -1)
  • slope of the line 8x + 5y + 13 = 0: m = -8/5
  • angle of inclination of line 8x + 5y + 13 = 0: 122.0°
  • points on the line 8x + 5y + 13 = 0: {(-5, 27/5), (-4, 19/5), (-3, 11/5), (-2, 3/5), (0, -13/5), (-13/8, 0), (1, -21/5) (2, -29/5), (3, -37/5), (4, -9), (5, -53/5)}
  • slope of the line 5x - 8y - 3 = 0: m = 5/8
  • angle of inclination of line 5x - 8y - 3 = 0: 32.0°
  • points on the line 5x - 8y - 3 = 0: {(-5, -7/2), (-4, -23/8), (-3, -9/4), (-2, -13/8), (0, -3/8), (3/5, 0), (1, 1/4), (2, 7/8), (3, 3/2), (4, 17/8), (5, 11/4))}


Find the point of intersection of lines x + 5y - 5 = 0 and 5x - y - 2 = 0.

graph of 2 perpendicular lines

  • point of intersection: (15/26, 23/26)
  • slope of the line x + 5y - 5 = 0: m = -1/5
  • angle of inclination of line x + 5y - 5 = 0: 168.7°
  • points on the line x + 5y - 5 = 0: {(-5, 2), (-4, 9/5), (-3, 8/5), (-2, 7/5), (-1, 6/5), (0, 1), (5, 0), (1, 4/5), (2, 3/5), (3, 2/5), (4, 1/5)}
  • slope of the line 5x - y - 2 = 0: m = 5
  • angle of inclination of line 5x - y - 2 = 0: 78.7°
  • points on the line 5x - y - 2 = 0: {(-5, -27), (-4, -22), (-3, -17), (-2, -12), (-1, -7), (0, -2), (2/5, 0), (1, 3), (2, 8), (3, 13), (4, 18), (5, 23)}


1-3 Graphs of Intersecting Non-Perpendicular Lines



Find the point of intersection of lines 5x + 6y - 8 = 0 and 2x + y + 1 = 0.

graph of 2 intersecting non-perpendicular lines
  • point of intersection: (-2, 3)
  • slope of the line 5x + 6y - 8 = 0: m = -5/6
  • angle of inclination of line 5x + 6y - 8 = 0: 140.2°
  • points on the line 5x + 6y - 8 = 0: {(-5, 11/2), (-4, 14/3), (-3, 23/6), (-1, 13/6), (0, 4/3), (8/5, 0), (1, 1/2), (2, -1/3), (3, -7/6), (4, -2), (5, -17/6)}
  • slope of the line 2x + y + 1 = 0: m = -2
  • angle of inclination of line 2x + y + 1 = 0: 116.6°
  • points on the line 2x + y + 1 = 0: {(-5, 9), (-4, 7), (-3, 5), (-1, 1), (0, -1), (-1/2, 0), (1, -3), (2, -5), (3, -7), (4, -9), (5, -11)}


Find the point of intersection of lines 5x + y + 16 = 0 and 3x - y + 8 = 0.

graph of 2 intersecting non-perpendicular lines

  • point of intersection: (-3, -1)
  • slope of the line 5x + y + 16 = 0: m = -5
  • angle of inclination of line 5x + y + 16 = 0: 101.3°
  • points on the line 5x + y + 16 = 0: {(-5, 9), (-4, 4), (-2, -6), (-1, -11), (0, -16), (-16/5, 0), (1, -21), (2, -26), (3, -31), (4, -36), (5, -41)}
  • slope of the line 3x - y + 8 = 0: m = 3
  • angle of inclination of line 3x - y + 8 = 0: 71.6°
  • points on the line 3x - y + 8 = 0: {(-5, -7), (-4, -4), (-2, 2), (-1, 5), (0, 8), (-8/3, 0), (1, 11), (2, 14), (3, 17), (4, 20), (5, 23)}


2 Graphs of Parallel Lines



Show that lines 3x + 5y - 4 = 0 and 3x + 5y + 15 = 0 are parallel.

graph of 2 parallel lines

  • points on the line 3x + 5y - 4 = 0: {(-5, 19/5), (-4, 16/5), (-3, 13/5), (-2, 2), (-1, 7/5), (0, 4/5), (4/3, 0), (1, 1/5), (2, -2/5), (3, -1), (4, -8/5), (5, -11/5)}
  • slope of the lines: m = -3/5
  • angle of inclination of the lines: 149.0°
  • points on the line 3x + 5y + 15 = 0: {(-5, 0), (-4, -3/5), (-3, -6/5), (-2, -9/5), (-1, -12/5), (0, -3), (1, -18/5), (2, -21/5), (3, -24/5), (4, -27/5), (5, -6)}


Show that lines 4x - 3y + 1 = 0 and 4x - 3y - 3 = 0 are parallel.

graph of 2 parallel lines

  • points on the line 4x - 3y + 1 = 0: {(-5, -19/3), (-4, -5), (-3, -11/3), (-2, -7/3), (-1, -1), (0, 1/3), (-1/4, 0), (1, 5/3), (2, 3), (3, 13/3), (4, 17/3), (5, 7)}
  • slope of the lines: m = 4/3
  • angle of inclination of the lines: 53.1°
  • points on the line 4x - 3y - 3 = 0: {(-5, -23/3), (-4, -19/3), (-3, -5), (-2, -11/3), (-1, -7/3), (0, -1), (3/4, 0), (1, 1/3), (2, 5/3), (3, 3), (4, 13/3), (5, 17/3)}
Posted by Gerardo Emerzona I

May 25, 2010

Algebra: Root of a Linear Equation With One Unknown (Non-Fractional



1. 4x + 5 = 7x - 10
Answer: x = 5
2. 3x - 4 = 8x + 41
Answer: x = -9
3. 11x + 7 = 122 - 12x
Answer: x = 5
4. 9x + 7 = 187 - 9x
Answer: x = 10
5. 10x + 7 = 7x - 26
Answer: x = -11
6. x + 3 = 7x + 21
Answer: x = -3
7. 8x + 7 = 37 - 2x
Answer: x = 3
8. 3x + 11 = 9x - 13
Answer: x = 4
9. 2x + 3 = 58 - 3x
Answer: x = 11
10. 11x + 5 = 29 + 7x
Answer: x = 6
11. 4x - 6 = 7 - 13x
Answer: x = 13/17
12. x - 4 = 5 - 5x
Answer: x = 3/2
13. 14x + 5 = 5x + 9
Answer: x = 4/9
14. 11x - 2 = 8x - 4
Answer: x = -2/3
15. 6x + 2 = 7 - 14x
Answer: x = 1/4
16. 4x - 13 = 14 - 11x
Answer: x = 9/5
17. 4x - 3 = 4 - 10x
Answer: x = 1/2
18. 3x + 10 = 15 - 12x
Answer: x = 1/3
19. 5x + 7 = 10 - 10x
Answer: x = 1/5
20. 11x - 1 = 2 - 13x
Answer: x = 1/8
Posted by Gerardo Emerzona I

May 9, 2010

Algebra: Graphing of Linear Equation Problems













Posted by Gerardo Emerzona I

April 22, 2010

Algebra: The Polynomial Arithmetic



The following sets of problems are about:

  • Addition of Polynomials
  • Subtraction of Polynomials
  • Multiplication of Polynomials
  • Division of Polynomials

Addition of Polynomials

1.9x - y - 2z and 7x + 5y - 8z
Answer: 16x + 4y - 10z
2.8m - 7n + 7 and 5m + 3n - 2
Answer: 13m - 4n + 5
3.5a + 3b + 9 and 2a - 9b + 7
Answer: 7a - 6b + 16
4.9a + 4b + 9c and 3a + 3b + 2c
Answer: 12a + 7b + 11c
5.6x - 7y - 6z, 9x + 8y - 7z and 9x - 5y - 7z
Answer: 24x - 4y - 20z
6.2x - 9y + 2z, 6x - 5y + z and 7x + 6y + 9
Answer: 15x - 8y + 3z + 9
7.5a + 6b - 3, 1a + b - 6c and 3a - 7b + 5
Answer: 9a - 6c + 2
8.6x - 7y - 6z, 9x + 8y - 7z and 9x - 5y - 7z
Answer: 24x - 4y - 20z
9.2x - 9y + 2z, 6x - 5y + z and 7x + 6y + 9
Answer: 15x - 8y + 3z + 9
10.3a + 5b + 9, 8a - 9b - 7c and 9a + b - 3
Answer: 20a - 3b - 7c + 6
11.6a2 - 6ab - 12a and 7a2 - 28ab - 7a
Answer: 13a2 - 34ab - 19a
12.14a2b + 56ab2 + 21ab and 42a2b - 7ab2 - 7ab
Answer: 56a2b + 49ab2 + 14ab
13.3a5 + 5a4b + 8a4 and 20a5 + 35a4b - 10a4
Answer: 23a5 + 40a4b - 2a4
14.35x2 + 28xy + 21x and 20x2 + 4xy - 28x
Answer: 55x2 + 32xy - 7x
15.10x2y + 5xy2 + 40xy and 49x2y + 14xy2 + 7xy
Answer: 59x2y + 19xy2 + 47xy
16.49x5 - 35x4y + 49x4 and 8x5 + 14x4y + 14x4
Answer: 57x5 - 21x4y + 63x4
17.4x4 + x3 + 9 and 2x4 + 7x3 + 3
Answer: 6x4 + 8x3 + 12
18.8m3 + m2 - 6 and 7m3 + 8m2 - 7
Answer: 15m3 + 9m2 - 13
19.4b2 + 9b + 5 and 2b2 + 5b + 2
Answer: 6b2 + 14b + 7
20.63x2 - 9xy - 63x and 30x2 - 40xy - 45x
Answer: 93x2 - 49xy - 108x


Subtraction of Polynomials

1.9x + 2y + 3z and 7x + 6y + 8z
Answer: 2x - 4y - 5z
2.2m + 8n + 2 and m - 7n - 8
Answer: m + 15n + 10
3.8a + 2b - 1 and 7a - b - 3
Answer: a + 3b + 2
4.6a + 5b + 8c and 4a - 7b + 7c
Answer: 2a + 12b + c
5.6x - 3y + 8z, x - 3y - 5z and 2x + 7y - 2z
Answer: 3x - 7y + 15z
6.9x + 4y + 9z, 4x - 7y + z and x + 5y - 2
Answer: 4x + 6y + 8z + 2
7.9a + 3b - 5, 8a + 7b + 4c and a - b + 4
Answer: - 3b - 4c - 9
8.8x - 4y + 5z, 3x + 4y - 7z and 5x + 4y - z
Answer: -12y + 13z
9.9x - 5y - 9z, 2x + 2y - 3z and 5x + 6y - 5
Answer: 2x - 13y - 6z + 5
10.9a + 3b - 8, a + 5b + 2c and 7a - 8b + 3
Answer: a + 6b - 2c - 11
11.4a2 - 20ab - 4a and a2 - 8ab + 3a
Answer: 3a2 - 12ab - 7a
12.63a2b + 36ab2 - 45ab and 9a2b - ab2 - 5ab
Answer: 54a2b + 37ab2 - 40ab
13.24a5 - 21a4b - 24a4 and 20a5 + 35a4b - 45a4
Answer: 4a5 - 56a4b + 21a4
14.24x2 + 24xy + 9x and 12x2 + 24xy - 4x
Answer: 12x2 + 13x
15.21x2y + 42xy2 + 7xy and 9x2y + 9xy2 + 63xy
Answer: 12x2y + 33xy2 - 56xy
16.18x3 - 63x2y - 81x2 and 6x3 - 5x2y - 8x2
Answer: 12x3 - 58x2y - 73x2
17.9x2 + 2x + 7 and 4x2 + 3x + 8
Answer: 5x2 - x - 1
18.9m2 - 7m + 6 and 8m2 - 2m + 1
Answer: m2 - 5m + 5
19.8b2 - 3b + 1 and 7b2 - b + 5
Answer: b2 - 2b - 4
20.5x2 - 7x + 8 and 4x2 - 5x + 7
Answer: x2 - 2x + 1


Multiplication of Polynomials

1.x - 3y and x - 7y
Answer: x2 - 10xy + 21y2
2.4a - 7b and 4a - 7b
Answer: 16a2 - 56ab + 49b2
3.6a + b and 6a + b
Answer: 36a2 + 12ab + b2
4.8x + y and 9x + 4y
Answer: 72x2 + 41xy + 4y2
5.2x - 7y and 2x + 7y
Answer: 4x2 - 49y2
6.3a - 4b and 9a + 2b
Answer: 27a2 - 30ab - 8b2
7.6mx + ny and 8mx - ny
Answer: 48m2x2 + 2mxny - n2y2
8.7ax + by and 8ax + 7by
Answer: 56a2x2 + 57axby + 7b2y2
9.(x - 5y - 6)2
Answer: x2 - 10xy + 25y2 - 12x + 60y + 36
10.(8p - 7q + 1)2
Answer: 64p2 - 112pq + 49q2 + 16p - 14q + 1
11.(8p - 7q + 4)2
Answer: 64p2 - 112pq + 49q2 + 64p - 56q + 16
12.(a - 5b - 6y)2
Answer: a2 - 10ab + 25b2 - 12a + 60b + 36y2
13.(5x4)(x + y)
Answer: 5x5 + 5x4y
14.(4x)(7x4 - 8y2)
Answer: 28x5 - 32xy2
15.(20xy)(3x4 - 8)
Answer: 60x5y - 160xy
16.(3ab4)(3a4 + 7)
Answer: 9a5b4 + 21ab4
17.(14s3t2)(3s3 - 7u2 + 7)
Answer: 42s6t2 - 98s3t2u2 + 98s3t2
18.(s + t)(u3 + v4)
Answer: su3 + sv4 + tu3 + tv4
19.(w + 4x)(5y + z)
Answer: 5wy + wz + 20xy + 4xz
20.(5s + 7t)(7u - 4v)
Answer: 35su - 20sv + 49tu - 28tv
21.(3w3 + 8x4)(2y2 + z3)
Answer: 6w3y2 + 3w3z3 + 16x4y2 + 8x4z3
22.(3w + 4x2)(4y2 - 3z4)
Answer: 12wy2 - 9wz4 + 16x2y2 - 12x2z4
23.(7a + 9b)(2b + 3)
Answer: 14ab + 21a + 18b2 + 27b
24.(x + 9y)(7y - 4)
Answer: 7xy - 4x + 63y2 - 36y
25.(8s3 + t)(4t4 - 7)
Answer: 32s3t4 - 56s3 + 4t5 - 7t
26.(s - t)(u + 8v)
Answer: su + 8sv - tu - 8tv
27.(x + 6)(x + 7)
Answer: x2 + 13x + 42
28.(x - 7)(x + 4)
Answer: x2 - 3x - 28
29.(x - 9)(x - 1)
Answer: x2 - 10x + 9
30.(8x + 1)(3x + 2)
Answer: 24x2 + 19x + 2
31.(x - 5)(4x + 3)
Answer: 4x2 - 17x - 15
32.(9x - 1)(5x - 2)
Answer: 45x2 - 23x + 2
33.(x + 9)(x + 9)
Answer: x2 + 18x + 81
34.(3x + 5)(3x + 5)
Answer: 9x2 + 30x + 25
35.(x - 1)(x - 1)
Answer: x2 - 2x + 1
36.(9x - 1)(9x - 1)
Answer: 81x2 - 18x + 1
37.[ 9x + 8(y - z) ] [ (9x - 8(y - z) ]
Answer: 81x2 - 64(y - z)2
38.(x + 2y) (x - 2y) (x2 + 4y2)
Answer: x4 - 16y4
39.(x - 8) (x2 + 8x + 64)
Answer: x3 - 512
40.(x2 - 7) (x4 + 7x2 + 49)
Answer: x6 - 343
41.(7y - 8) (49y2 + 56y + 64)
Answer: 343y3 - 512
42.(2y2 - 9) (4y4 + 18y2 + 81)
Answer: 8y6 - 729
43.(3a - 4b) (9a2 + 12ab + 16b2)
Answer: 27a3 - 64b3
44.(3a2 - 7b) (9a4 + 21a2b + 49b2)
Answer: 27a6 - 343b3
45.(9xy2 - 4z) (81x2y4 + 36xy2z + 16z2)
Answer: 729x3y6 - 64z3
46.x (2s + 9t)(4s2 - 18st + 81t2)
Answer: 8s3x + 729t3x
47.x (8s2 + 2t2)(64s4 - 16s2t2 + 4t4)
Answer: 512s6x + 8t6x
48.(8x + 5y) (64x2 - 40xy + 25y2)
Answer: 512x3 + 125y3
49.(7x2 + 9y2) (49x4 - 63x2y2 + 81y4)
Answer: 343x6 + 729y6
50.4x9 (y + 7z) (y - 7z)
Answer: 4x9y2 - 196x9z2


Division of Polynomials

1.63a4b7x9 ÷ 9a3b5x4
Answer: 7ab2x5
2.9x7y6z4 ÷ x2y3z3
Answer: 9x5y3z
3.-14m4n7 ÷ 2m3n4
Answer: -7mn3
4.72x2y2 ÷ 8xy
Answer: 9xy
5.24a6b5c7 ÷ 3a2b4c3
Answer: 8a4bc4
6.28x5y2z + 63x3y3z + 49x2y3z ÷ 7x2y2z
Answer: 4x3 + 9xy + 7y
7.x2 - 8x - 209 ÷ x - 19
Answer: x + 11
8.x2 - 29x + 204 ÷ x - 9
Answer: x - 20; remainder = 24
9.x2 - 22x + 88 ÷ x - 4
Answer: x - 18; remainder = 16
10.x4 + 3x2 - 460 ÷ x2 + 23
Answer: x2 - 20
11.x4 - 21x2 + 130 ÷ x2 - 10
Answer: x2 - 11; remainder = 20
12.x4 - 6x3y - 25x2y2 + 52xy3 + 14y4 ÷ x2 + 2xy - 7y2
Answer: x2 - 8xy - 2y2
13.x4 - 8x3y + 7x2y2 + 34xy3 + 15y4 ÷ x2 - 5xy - 3y2
Answer: x2 - 3xy - 5y2
14.8x3 - 57x2 + 48x + 42 ÷ x - 6
Answer: 8x2 - 9x - 6; remainder = 6
15.56x5 - 111x4 + 94x3 - 70x2 + 27x - 47 ÷ 7x3 - 6x2 - 2x - 5
Answer: 8x2 - 9x + 8; remainder = -2x - 7
16.6a5x4y9 ÷ a3x6y6
Answer: 6a2x-2y3
17.56p9q2r4 ÷ 8p3q6r5
Answer: 7p6q-4r-1
18.63x5 + 88x4 + 41x3 - 63x - 63x2 + 9 ÷ 7x3 + 9x2 + 9x - 1
Answer: 9x2 + x - 7; remainder = x + 2
19.32x3 + 44x2 + 25x + 21 ÷ 8x + 9
Answer: 4x2 + x + 2; remainder = 3
20.x4 + 11x3y + 25x2y2 - 7xy3 - 6y4 ÷ x2 + 8xy + 3y2
Answer: x2 + 3xy - 2y2
Posted by Gerardo Emerzona I
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