Analytic Geometry: Equation and Slope of a Line

Given two points each for line P1P2 and line AB, the equations and slopes of the lines are determined. Through their slopes,

it's determined whether the lines are parallel or perpendicular to each other.

Problem	Given Points	Equations of the Line	Slopes of the Line
1	P1(0,8) P2(2,3); A(11,3) B(9,8)	5x+2y-16=0, 5x+2y-61=0	parallel lines: m1=-5/2 m2=5/-2
2	P1(7,0) P2(0,8); A(1,0) B(9,7)	8x+7y-56=0, 7x-8y-7=0	perpendicular lines: m1=8/-7 m2=7/8
3	P1(5,5) P2(1,0); A(3,8) B(8,4)	5x-4y-5=0, 4x+5y-52=0	perpendicular lines: m1=-5/-4 m2=-4/5
4	P1(-4,7) P2(0,8); A(11,7) B(7,6)	x-4y+32=0, x-4y+17=0	parallel lines: m1=1/4 m2=-1/-4
5	P1(-8,0) P2(2,9); A(4,2) B(14,11)	9x-10y+72=0, 9x-10y-16=0	parallel lines: m1=9/10 m2=9/10
6	P1(9,9) P2(8,2); A(9,6) B(2,7)	7x-y-54=0, x+7y-51=0	perpendicular lines: m1=-7/-1 m2=1/-7
7	P1(9,5) P2(8,1); A(8,7) B(4,8)	4x-y-31=0, x+4y-36=0	perpendicular lines: m1=-4/-1 m2=1/-4
8	P1(-7,0) P2(4,8); A(12,12) B(1,4)	8x-11y+56=0, 8x-11y+36=0	parallel lines: m1=8/11 m2=-8/-11
9	P1(-8,0) P2(4,6); A(4,6) B(14,11)	x-2y+8=0, x-2y+8=0	parallel lines: m1=1/2 m2=1/2
10	P1(-9,5) P2(9,8); A(10,2) B(4,1)	x-6y+39=0, x-6y+2=0	parallel lines: m1=1/6 m2=-1/-6
11	P1(3,1) P2(2,3); A(6,7) B(8,8)	2x+y-7=0, x-2y+8=0	perpendicular lines: m1=2/-1 m2=1/2
12	P1(2,3) P2(1,1); A(2,6) B(4,5)	2x-y-1=0, x+2y-14=0	perpendicular lines: m1=-2/-1 m2=-1/2
13	P1(-2,1) P2(2,7); A(12,13) B(10,10)	3x-2y+8=0, 3x-2y-10=0	parallel lines: m1=3/2 m2=-3/-2
14	P1(9,1) P2(8,3); A(4,7) B(6,8)	2x+y-19=0, x-2y+10=0	perpendicular lines: m1=2/-1 m2=1/2
15	P1(8,3) P2(4,0); A(1,4) B(4,0)	3x-4y-12=0, 4x+3y-16=0	perpendicular lines: m1=-3/-4 m2=-4/3
16	P1(-8,1) P2(2,6); A(8,3) B(2,0)	x-2y+10=0, x-2y-2=0	parallel lines: m1=1/2 m2=-1/-2
17	P1(-4,4) P2(7,8); A(1,7) B(12,11)	4x-11y+60=0, 4x-11y+73=0	parallel lines: m1=4/11 m2=4/11
18	P1(4,6) P2(1,1); A(4,6) B(9,3)	5x-3y-2=0, 3x+5y-42=0	perpendicular lines: m1=-5/-3 m2=-3/5
19	P1(4,1) P2(5,3); A(4,5) B(2,6)	2x-y-7=0, x+2y-14=0	perpendicular lines: m1=2/1 m2=1/-2
20	P1(-5,9) P2(4,4); A(4,13) B(13,8)	5x+9y-56=0, 5x+9y-137=0	parallel lines: m1=-5/9 m2=-5/9

Algebra: Linear Equations in Two Unknown

Given two linear equations, the two equations are solved simultaneously for x and y.

Problem	Given Equations	Solution
1	6x+3y = 45; 8x+4y = 60	x = 14; y = -13
2	3x-9y = -114; 7x+4y = 84	x = 4; y = 14
3	4x+8y = 104; 5x+y = 76	x = 14; y = 6
4	4x+3y = 59; 2x+9y = 97	x = 8; y = 9
5	6x+y = 21; 2x+5y = 49	x = 2; y = 9
6	3x+3y = 30; x+3y = 14	x = 8; y = 2
7	5x+7y = 139; 4x+5y = 104	x = 11; y = 12
8	6x+7y = 125; 7x+4y = 100	x = 8; y = 11
9	5x+2y = 60; x+8y = 50	x = 10; y = 5
10	9x+y = 16; 4x+9y = -10	x = 2; y = -2
11	x+8y = 63; 6x+4y = 70	x = 7; y = 7
12	8x+4y = 44; 6x+y = 31	x = 5; y = 1
13	2x+6y = 96; 3x+6y = 102	x = 6; y = 14
14	7x+4y = 63; 5x+8y = 117	x = 1; y = 14
15	8x+5y = 102; 6x+8y = 102	x = 9; y = 6
16	6x+y = 11; 5x+7y = 40	x = 1; y = 5
17	x+8y = 41; 8x+6y = 38	x = 1; y = 5
18	x+2y = 0; 5x+7y = -12	x = -8; y = 4
19	6x+4y = 20; 6x+4y = 20	x = 2; y = 2
20	x+3y = 22; 6x+3y = 42	x = 4; y = 6
21	5x+5y = 95; x+2y = 29	x = 9; y = 10
22	5x-8y = 52; 8x+y = 97	x = 12; y = 1
23	9x+9y = 234; 8x+y = 124	x = 14; y = 12
24	9x+6y = 93; 8x+6y = 88	x = 5; y = 8
25	7x+4y = 41; 5x+7y = -8	x = 11; y = -9
26	2x+6y = -48; 7x+9y = -24	x = 12; y = -12
27	3x+3y = 33; 7x+9y = 89	x = 5; y = 6
28	3x+9y = 96; 3x+9y = 96	x = 5; y = 9
29	9x+2y = 62; 5x+3y = 59	x = 4; y = 13
30	x+6y = 72; 7x+8y = 164	x = 12; y = 10
31	7x+4y = 43; 8x+4y = 44	x = 1; y = 9
32	4x+6y = 88; 3x-5y = -29	x = 7; y = 10
33	7x+4y = 18; 6x+3y = 15	x = 2; y = 1
34	7x+2y = 32; 8x+8y = 88	x = 2; y = 9
35	x+3y = 53; 4x+3y = 95	x = 14; y = 13
36	6x+9y = 87; 2x+6y = 38	x = 10; y = 3
37	3x+9y = 36; 4x+2y = 18	x = 3; y = 3
38	4x+5y = 93; 6x+6y = 120	x = 7; y = 13
39	2x+5y = 51; 8x+5y = 69	x = 3; y = 9
40	3x+9y = 126; 5x+9y = 138	x = 6; y = 12
41	8x+7y = 124; 2x+7y = 94	x = 5; y = 12
42	3x+2y = 2; 7x+6y = -10	x = 8; y = -11
43	9x+6y = 135; 8x+7y = 140	x = 7; y = 12
44	7x+y = 34; 4x+2y = 28	x = 4; y = 6
45	9x+6y = 33; 8x+7y = 46	x = -3; y = 10
46	8x+y = 54; 3x+5y = -26	x = 8; y = -10
47	4x+7y = 116; 5x+7y = 124	x = 8; y = 12
48	2x+4y = 38; 2x+5y = 42	x = 11; y = 4
49	5x+5y = 75; 6x-8y = -64	x = 4; y = 11
50	9x+8y = 126; 6x+2y = 54	x = 6; y = 9

Addition of Fractions

Problem	First Number	Second Number	Total
1	7/24	7/18	49/72
2	1/20	5/18	59/180
3	7/8	1/24	11/12
4	7/24	7/22	161/264
5	6/7	5/8	1 27/56
6	7/20	2/15	29/60
7	3/5	8/9	1 22/45
8	3/14	1/20	37/140
9	1/20	4/15	19/60
10	1/9	11/24	41/72
11	1/24	5/21	47/168
12	3/5	7/10	1 3/10
13	4/21	5/16	169/336
14	7/18	5/18	2/3
15	5/22	7/10	51/55
16	6/7	4/7	1 3/7
17	4/21	3/10	103/210
18	1/15	1/15	2/15
19	7/18	7/16	119/144
20	5/14	7/12	79/84
21	5/12	1/10	31/60
22	7/24	8/21	113/168
23	5/18	7/16	103/144
24	8/9	4/9	1 1/3
25	5/24	1/20	31/120
26	3/16	5/12	29/48
27	1/14	5/18	22/63
28	1/14	7/20	59/140
29	8/21	8/21	16/21
30	7/15	3/14	143/210
31	7/12	7/18	35/36
32	7/12	7/24	7/8
33	1/10	1/14	6/35
34	7/18	3/22	52/99
35	5/16	1/20	29/80
36	8/21	5/24	33/56
37	7/18	7/20	133/180
38	1/14	3/14	2/7
39	6/7	1/7	1
40	8/15	7/18	83/90
41	5/24	5/18	35/72
42	3/4	14/15	1 41/60
43	4/5	7/8	1 27/40
44	1/16	4/21	85/336
45	1/21	2/21	1/7
46	5/16	5/14	75/112
47	5/24	7/20	67/120
48	3/10	7/18	31/45
49	8/9	4/9	1 1/3
50	1/16	7/24	17/48

Subtraction of Fractions

Problem	Minuend	Subtrahend	Difference
1	7/15	3/10	1/6
2	1/18	1/16	-1/144
3	5/21	7/15	-8/35
4	1/18	1/10	-2/45
5	5/14	1/14	2/7
6	6/7	1/7	5/7
7	1/16	7/15	-97/240
8	7/10	5/16	31/80
9	7/10	1/24	79/120
10	6/7	8/9	-2/63
11	1/14	5/24	-23/168
12	6/7	2/7	4/7
13	7/12	1/12	1/2
14	7/20	7/12	-7/30
15	3/16	8/21	-65/336
16	7/8	6/7	1/56
17	1/21	1/14	-1/42
18	1/12	7/12	-1/2
19	2/7	3/16	11/112
20	1/3	11/12	-7/12
21	2/15	8/15	-2/5
22	1/24	1/10	-7/120
23	3/20	5/14	-29/140
24	3/14	5/18	-4/63
25	5/12	7/10	-17/60
26	7/24	2/21	11/56
27	6/7	1/15	83/105
28	3/10	7/16	-11/80
29	7/24	7/24	0
30	7/10	5/12	17/60
31	7/22	5/21	37/462
32	4/21	7/18	-25/126
33	3/4	5/4	-1/2
34	7/24	1/10	23/120
35	7/9	7/18	7/18
36	7/24	3/22	41/264
37	5/18	7/24	-1/72
38	6/7	11/15	13/105
39	7/16	5/18	23/144
40	7/16	5/14	9/112
41	5/18	4/15	1/90
42	7/24	4/21	17/168
43	3/14	7/22	-8/77
44	1/15	1/14	-1/210
45	7/16	1/16	3/8
46	3/16	7/18	-29/144
47	7/24	7/18	-7/72
48	7/24	5/24	1/12
49	7/18	3/14	11/63
50	8/9	7/24	43/72

Multiplication of Fractions

Problem	Multiplicand	Multiplier	Product
1	3/7	2/7	6/49
2	1/20	7/10	7/200
3	2/5	1/20	1/50
4	7/22	3/10	21/220
5	1/12	2/15	1/90
6	7/18	1/10	7/180
7	8/9	1/7	8/63
8	7/24	7/18	49/432
9	3/10	7/10	21/100
10	1/16	3/20	3/320
11	1/7	6/7	6/49
12	3/22	7/16	21/352
13	7/15	3/14	1/10
14	7/22	7/15	49/330
15	7/22	1/24	7/528
16	1/3	1/12	1/36
17	3/20	5/22	3/88
18	5/24	7/20	7/96
19	1/15	7/20	7/300
20	7/22	7/12	49/264
21	5/12	5/16	25/192
22	3/4	9/14	27/56
23	2/15	7/12	7/90
24	3/14	1/22	3/308
25	7/20	7/10	49/200
26	3/10	7/16	21/160
27	3/22	3/22	9/484
28	8/9	7/8	7/9
29	5/12	7/20	7/48
30	5/12	3/14	5/56
31	1/16	5/12	5/192
32	7/9	5/9	35/81
33	2/21	3/14	1/49
34	7/18	1/14	1/36
35	8/9	7/18	28/81
36	1/20	7/22	7/440
37	5/18	1/15	1/54
38	1/18	7/10	7/180
39	1/18	3/10	1/60
40	8/9	10/21	80/189
41	8/9	4/9	32/81
42	4/15	1/16	1/60
43	3/22	7/20	21/440
44	7/16	7/22	49/352
45	1/10	5/12	1/24
46	1/15	2/15	2/225
47	5/9	9/22	5/22
48	5/24	5/14	25/336
49	1/22	1/12	1/264
50	5/7	5/8	25/56