Systems of Linear Equations In Three Unknowns

These problems are solved by converting the given system into a system of equations in two unknowns.

I kept the variables to minimum in order to avoid having to compute very large numbers that may distract the students from understanding the methods involved in solving this particular type of algebraic problem.

1.	3x - 5y + 4z = 23 and x - 2y + 2z = 9 and 5x - 6y - 3z = 14
	Answer: x = 7; y = 2; z = 3
2.	5x + 2y - 5z = 0 and 6x + y - 2z = 9 and 5x - 4y - 5z = -30
	Answer: x = 2; y = 5; z = 4
3.	5x - y - z = 14 and 5x - 2y - 3z = 7 and 3x - 2y + 3z = 5
	Answer: x = 4; y = 5; z = 1
4.	4x - 3y + 2z = -1 and 3x + y - 3z = 4 and 6x - y - 4z = -5
	Answer: x = 3; y = 7; z = 4
5.	3x + 6y + z = 11 and 2x - 5y - 5z = -13 and x + 4y + 3z = 11
	Answer: x = 1; y = 1; z = 2
6.	3x + y + 2z = 1 and 3x - y + 6z = -1 and 3x - 2y + 3z = 13
	Answer: x = 4; y = -5; z = -3
7.	5x - 2y - 2z = -8 and x + 6y - z = 0 and 5x + 4y + 2z = 30
	Answer: x = 2; y = 1; z = 8
8.	4x - 3y + z = -14 and 6x + y + 4z = 3 and 2x + 3y + 4z = 17
	Answer: x = -2; y = 3; z = 3
9.	3x - y - z = -9 and 2x + y + z = -1 and x - 5y - 3z = -9
	Answer: x = -2; y = -1; z = 4
10.	2x - y - 3z = -11 and x + 3y + z = 14 and x - 3y - 2z = -13
	Answer: x = 3; y = 2; z = 5
11.	3x + 4y + 4z = 34 and 2x - 3y + 2z = 15 and 2x + y - 3z = 4
	Answer: x = 6; y = 1; z = 3
12.	3x - y + 4z = 6 and x + 3y + z = 15 and 3x - 4y - 3z = -13
	Answer: x = 2; y = 4; z = 1
13.	3x + 3y + 2z = -8 and 3x - 2y - 5z = -8 and 5x + 4y + z = -18
	Answer: x = 1; y = -7; z = 5
14.	x - y + 4z = 4 and 5x + y + 3z = -11 and 5x + 3y + z = -3
	Answer: x = -7; y = 9; z = 5
15.	3x - 2y + 3z = -15 and 5x + 4y + z = -11 and x + 5y + 3z = 16
	Answer: x = -5; y = 3; z = 2
16.	3x + 5y - 2z = 14 and 5x + 3y - 4z = 16 and 3x - 4y + 3z = 20
	Answer: x = 5; y = 1; z = 3
17.	6x + y + 3z = -6 and 7x - 4y + 5z = -2 and x - 6y + 5z = -14
	Answer: x = 3; y = -3; z = -7
18.	6x - y - 6z = 23 and 5x - 4y + z = -20 and x - 6y - 5z = 22
	Answer: x = -2; y = 1; z = -6
19.	5x + 3y - 2z = 15 and 3x + 2y + 3z = 27 and 3x + 5y - 5z = 13
	Answer: x = 1; y = 6; z = 4
20.	4x - 3y + 2z = -2 and 3x - y + 3z = 8 and 6x - y + 4z = 14
	Answer: x = 1; y = 4; z = 3
21.	x - y - 4z = -22 and 5x - y - 3z = 11 and 5x - 3y - z = 3
	Answer: x = 7; y = 9; z = 5
22.	4x - 3y + 2z = 11 and 2x + 3y + z = 10 and 4x + 3y + 4z = 23
	Answer: x = 2; y = 1; z = 3
23.	4x + 3y + 2z = 27 and x + 2y + 3z = 13 and 2x + 3y + z = 18
	Answer: x = 4; y = 3; z = 1
24.	x - 3y - 4z = -33 and x + y - 5z = -38 and x - 5y - z = -8
	Answer: x = 6; y = 1; z = 9
25.	5x - 3y + 5z = -15 and 3x + y + 5z = -1 and 3x - y - 5z = 19
	Answer: x = 3; y = 5; z = -3
26.	x - y - 7z = -8 and 9x - 4y - 5z = 11 and 5x - y + 3z = 18
	Answer: x = 4; y = 5; z = 1
27.	2x - y - 5z = -13 and 2x - 3y - 5z = -17 and x - 2y - 2z = -8
	Answer: x = 2; y = 2; z = 3
28.	x - 3y + 3z = 3 and 2x + y + 2z = 9 and x - y + 2z = 4
	Answer: x = 3; y = 1; z = 1
29.	x - 3y + 6z = 25 and x - 4y + 9z = 38 and x - y + 5z = 24
	Answer: x = 1; y = 2; z = 5
30.	x - 3y + z = -10 and 3x + 3y - 4z = 21 and 2x - y - 3z = -5
	Answer: x = 5; y = 6; z = 3

Coordinate Graphs of Triangles

The illustrated problems below are graphs of the following triangles:

• isosceles triangles
• scalene triangles
• right triangles

Concepts Illustrated By the Problems

• distance between two points
• angle between two lines
• point of intersection of two straight lines
• equation of a straight line

Description of the Problems

For each problem, the following information are given:

• equations of the line segments
• slopes of the line segments
• x-intercepts and y-intercepts of the lines
• distances between the terminal points of the line segments
• interior angles of the triangles

Using one or a combination of two or more of the above information, it is possible for a problem to be presented in several ways.

Line segments and their terminal points referred to in the problems are the sides and vertices, respectively, of the triangles.

Slope of a line segment with terminal points A and B is abbreviated as m_AB.

Distance between points A and B is abbreviated as distance_AB

Graphs of Isosceles Triangles

Show that points A(-5, 1), B(3, 5), C(2, -3) are vertices of an isosceles triangle.



Distance Between Two Points

• distance_AB = √ 80
• distance_BC = distance_AC = √ 65

Equations of the Line Segments

• line segment AB: x - 2y + 7 = 0 (x-intercept = -7; y-intercept = 7/2)
• line segment BC: 8x - y - 19 = 0 (x-intercept = 19/8, y-intercept = -19)
• line segment AC: 4x + 7y + 13 = 0 (x-intercept = - 13/4, y-intercept = - 13/7)

Slopes of the Line Segments

• m_AB = 1/2
• m_BC = 8
• m_AC = - 4/7

Interior Angles of the Triangle

• angle A = angle B = 56.3°
• angle C = 67.4°

Show that points A(-5, 4), B(3, 3), C(-1, -4) are vertices of an isosceles triangle.



Distance Between Two Points

• distance_AB = distance_BC = √ 65
• distance_AC = √ 80

Equations of the Line Segments

• line segment AB: x + 8y - 27 = 0 ( x-intercept = 27, y-intercept = 27/8)
• line segment BC: 7x - 4y - 9 = 0 (x-intercept = 9/7, y-intercept = - 9/4)
• line segment AC: 2x + y + 6 = 0 (x-intercept = -3, y-intercept = -6)

Slopes of the Line Segments

• m_AB = - 1/8
• m_BC = 7/4
• m_AC = -2

Interior Angles of the Triangle

• angle A = angle C = 56.3°
• angle B = 67.4°

Prove analytically that points A(-5, -5), B(-2, 6), C(5, -3) are vertices of an isosceles triangle.



Distance Between Two Points

• distance_AB = distance_BC = √ 130
• distance_AC = √ 104

Equations of the Line Segments

• line segment AB: 11x - 3y + 40 = 0 (x-intercept = - 40/11, y-intercept = 40/3)
• line segment BC: 9x + 7y - 24 = 0 (x-intercept = 24/9, y-intercept = 24/7)
• line segment AC: x - 5y - 20 = 0 (x-intercept = 20, y-intercept = -4)

Slopes of the Line Segments

• m_AB = 11/3
• m_BC = - 9/7
• m_AC = 1/5

Interior Angles of the Triangle

• angle A = angle C = 63.4°
• angle B = 53.1°

Graphs of Scalene Triangles

Show that points A(-4, 1), B(1, 0), C(-2, -5) are vertices of a scalene triangle.



Distance Between Two Points

• distance_AB = √ 26
• distance_BC = √ 34
• distance_AC = √ 40

Equations of the Line Segments

• line segment AB: x + 5y - 1 = 0 (x-intercept = 1, y-intercept = 1/5)
• line segment BC: 5x - 3y - 5 = 0 (x-intercept = 1, y-intercept = - 5/3)
• line segment AC: 3x + y + 11 = 0 (x-intercept = - 11/3, y-intercept = -11)

Slopes of the Line Segments

• m_AB = - 1/5
• m_BC = 5/3
• m_AC = -3

Interior Angles of the Triangle

• angle A = 60.3°
• angle B = 70.3°
• angle C = 49.4°

Show that the following points are vertices of a scalene triangle:

• A (-5, -4)
• B (3, 4)
• C (4, -2)

Distance Between Two Points

• distance_AB = √ 128
• distance_BC = √ 37
• distance_AC = √ 85

Equations of the Line Segments

• line segment AB: x - y + 1 = 0 (x-intercept = -1, y-intercept = 1)
• line segment BC: 6x + y - 22 = 0 (x-intercept = 11/3, y-intercept = 22)
• line segment AC: 2x - 9y - 26 = 0 (x-intercept = 13, y-intercept = - 26/9)

Slopes of the Line Segments

• m_AB = 1
• m_BC = -6
• m_AC = 2/9

Interior Angles of the Triangle

• angle A = 32.5°
• angle B = 54.5°
• angle C = 93.0°

Show that the points A(-3, 5), B(5, -2), C(-1, -5) are vertices of a scalene triangle.



Distance Between Two Points

• distance_AB = √ 113
• distance_BC = √ 45
• distance_AC = √ 104

Equations of the Line Segments

• line segment AB: 7x + 8y - 19 = 0 (x-intercept = 19/7, y-intercept = 19/8)
• line segment BC: x - 2y - 9 = 0 (x-intercept = 9, y-intercept = - 9/2)
• line segment AC: 5x + y + 10 = 0 (x-intercept = -2, y-intercept = -10)

Slopes of the Line Segments

• m_AB = - 7/8
• m_BC = 1/2
• m_AC = -5

Interior Angles of the Triangle

• angle A = 37.5°
• angle B = 67.8°
• angle C = 74.7°

Graphs of Right Triangles

Prove that points A(-1, 5), B(5, -1), C(-4, 2) are vertices of a right triangle.

Distance Between Two Points

• distance_AB = √ 72
• distance_BC = √ 90
• distance_AC = √ 18

Equations of the Line Segments

• line segment AB: x + y - 4 = 0 (x-intercept = 4, y-intercept = 4)
• line segment BC: x + 3y - 2 = 0 (x-intercept = 2, y-intercept = 2/3)
• line segment AC: x - y + 6 = 0 (x-intercept = -6, y-intercept = 6)

Slopes of the Line Segments

• m_AB = -1
• m_BC = - 1/3
• m_AC = 1

Interior Angles of the Triangle

• angle A = 90.0°
• angle B = 26.6°
• angle C = 63.4°

Show that points A(-5, 3), B(2, -1), C(-1, -3) are vertices of a right triangle.



Distance Between Two Points

• distance_AB = √ 65
• distance_BC = √ 13
• distance_AC = √ 52

Equations of the Line Segments

• line segment AB: 4x + 7y - 1 = 0 (x-intercept = 1/4, y-intercept = 1/7)
• line segment BC: 2x - 3y - 7 = 0 (x-intercept = 7/2, y-intercept = - 7/3)
• line segment AC: 3x + 2y + 9 = 0 (x-intercept = -3, y-intercept = - 9/2)

Slopes of the Line Segments

• m_AB = - 4/7
• m_BC = 2/3
• m_AC = - 3/2

Interior Angles of the Triangle

• angle A = 26.6°
• angle B = 63.4°
• angle C = 90.0°

Prove that points A(-2, 5), B(4, -3), C(0, -6) are vertices of a right triangle.



Distance Between Two Points

• distance_AB = √ 100
• distance_BC = √ 25
• distance_AC = √ 125

Equations of the Line Segments

• line segment AB: 4x + 3y - 7 = 0 (x-intercept = 7/4, y-intercept = 7/3)
• line segment BC: 3x - 4y - 24 = 0 (x-intercept = 8, y-intercept = -6)
• line segment AC: 11x + 2y + 12 = 0 (x-intercept = - 12/11, y-intercept = -6)

Slopes of the Line Segments

• m_AB = - 4/3
• m_BC = 3/4
• m_AC = - 11/2

Interior Angles of the Triangle

• angle A = 26.6°
• angle B = 90.0°
• angle C = 63.4°

Graphs of Isosceles Right Triangles

Prove that the points A(-4, 1), B(1, 5), C(0, -4) are vertices of an isosceles right triangle.



Distance Between Two Points

• distance_AB = √ 41
• distance_BC = √ 82
• distance_AC = √ 41

Equations of the Line Segments

• line segment AB: 4x - 5y + 21 = 0 (x-intercept = - 21/4, y-intercept = 21/5)
• line segment BC: 9x - y - 4 = 0 (x-intercept = 4/9, y-intercept = -4)
• line segment AC: 5x + 4y + 16 = 0 (x-intercept = - 16/5, y-intercept = -4)

Slopes of the Line Segments

• m_AB = 4/5
• m_BC = 9
• m_AC = - 5/4

Interior Angles of the Triangle

• angle A = 90°
• angle B = 45°
• angle C = 45°

Show that points A(-5, 4), B(0, 2), C(-2, -3) are vertices of an isosceles right triangle.



Distance Between Two Points

• distance_AB = √ 29
• distance_BC = √ 29
• distance_AC = √ 58

Equations of the Line Segments

• line segment AB: 2x + 5y - 10 = 0 (x-intercept = 5, y-intercept = 2)
• line segment BC: 5x - 2y + 4 = 0 (x-intercept = - 4/5, y-intercept = 2)
• line segment AC: 7x + 3y + 23 = 0 (x-intercept = - 23/7, y-intercept = - 23/3)

Slopes of the Line Segments

• m_AB = - 2/5
• m_BC = 5/2
• m_AC = - 7/3

Interior Angles of the Triangle

• angle A = 45°
• angle B = 90°
• angle C = 45°

Prove that points A(-3, 4), B(5, 1), C(-6, -4) are vertices of an isosceles right triangle.



Distance Between Two Points

• distance_AB = √ 73
• distance_BC = √ 146
• distance_AC = √ 73

Equations of the Line Segments

• line segment AB: 3x + 8y - 23 = 0 (x-intercept = 23/3, y-intercept = 23/8)
• line segment BC: 5x - 11y - 14 = 0 (x-intercept = 14/5, y-intercept = - 14/11)
• line segment AC: 8x - 3y + 36 = 0 (x-intercept = - 9/2, y-intercept = 12)

Slopes of the Line Segments

• m_AB = - 3/8
• m_BC = 5/11
• m_AC = 8/3

Interior Angles of the Triangle

• angle A = 90°
• angle B = 45°
• angle C = 45°

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Simultaneous Equations

1.	2x - 3y = -26 and 4x - 7y = -60
	Answer: x = -1; y = 8
2.	3x + 5y = 21 and 7x - 2y = -74
	Answer: x = -8; y = 9
3.	3x + 7y = -42 and 8x + 9y = -25
	Answer: x = 7; y = -9
4.	5x - 3y = 51 and 7x + 5y = 7
	Answer: x = 6; y = -7
5.	8x - 7y = 58 and 5x + 3y = 51
	Answer: x = 9; y = 2
6.	3x - 5y = 7 and 9x - 4y = -34
	Answer: x = -6; y = -5
7.	5x + 8y = -37 and 3x - 2y = 5
	Answer: x = -1; y = -4
8.	9x - 5y = -81 and 3x - 8y = -84
	Answer: x = -4; y = 9
9.	4x + 5y = 59 and 2x - 7y = -37
	Answer: x = 6; y = 7
10.	9x + 2y = 37 and x + 7y = 38
	Answer: x = 3; y = 5
11.	9x - 8y = -31 and 7x - 4y = -13
	Answer: x = 1; y = 5
12.	7x - 5y = -63 and 2x + 9y = 55
	Answer: x = -4; y = 7
13.	6x + 5y = 75 and 8x + 3y = 67
	Answer: x = 5; y = 9
14.	4x - 9y = 32 and 3x - 2y = 5
	Answer: x = -1; y = -4
15.	4x + 7y = -45 and 3x - 8y = 59
	Answer: x = 1; y = -7
16.	x + 8y = 19 and 5x + 2y = 19
	Answer: x = 3; y = 2
17.	5x + 3y = -15 and 6x + 5y = -11
	Answer: x = -6; y = 5
18.	3x - 4y = 19 and 9x + 5y = 91
	Answer: x = 9; y = 2
19.	2x - 3y = -14 and 4x - 7y = -36
	Answer: x = 5; y = 8
20.	5x + 7y = -69 and 3x + 2y = -26
	Answer: x = -4; y = -7
21.	8x + 7y = 79 and 2x - 9y = 9
	Answer: x = 9; y = 1
22.	9x - 7y = -12 and 9x - 5y = -6
	Answer: x = 1; y = 3
23.	7x - 6y = -7 and 5x + 4y = 53
	Answer: x = 5; y = 7
24.	7x - 2y = -37 and 6x - 7y = -37
	Answer: x = -5; y = 1
25.	7x - 8y = -10 and 3x + 2y = 12
	Answer: x = 2; y = 3
26.	6x + 5y = 53 and 2x - 7y = -43
	Answer: x = 3; y = 7
27.	3x - 8y = 61 and 6x + 5y = -46
	Answer: x = -1; y = -8
28.	4x - y = -3 and 5x + 4y = 33
	Answer: x = 1; y = 7
29.	9x + 7y = 47 and 5x + 2y = 28
	Answer: x = 6; y = -1
30.	9x + 8y = 49 and 7x + 4y = 27
	Answer: x = 1; y = 5