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+2y = 68; 9x+8y = 62	x = 14; y = -8
2	2x-4y = -38; 9x+4y = 93	x = 5; y = 12
3	2x+3y = 60; 9x+3y = 144	x = 12; y = 12
4	9x+3y = 72; x+8y = 77	x = 5; y = 9
5	8x+y = 43; 7x+5y = 83	x = 4; y = 11
6	3x+8y = 95; 9x+3y = 138	x = 13; y = 7
7	6x+6y = 18; 4x+3y = 11	x = 2; y = 1
8	4x+5y = 84; x+9y = 83	x = 11; y = 8
9	6x+y = 38; x+8y = 22	x = 6; y = 2
10	3x+3y = -18; 4x+8y = -60	x = 3; y = -9
11	7x+5y = 47; 4x+7y = 31	x = 6; y = 1
12	8x+7y = 154; 3x+9y = 96	x = 14; y = 6
13	4x+3y = 70; 7x+8y = 161	x = 7; y = 14
14	6x+y = 63; 5x+6y = 99	x = 9; y = 9
15	3x+2y = 61; 2x+2y = 50	x = 11; y = 14
16	7x+4y = 90; 6x+7y = 120	x = 6; y = 12
17	2x+5y = 84; 7x+2y = 77	x = 7; y = 14
18	5x+8y = 23; 8x+5y = -10	x = -5; y = 6
19	6x+5y = 102; 8x+y = 102	x = 12; y = 6
20	x+9y = 20; 2x+7y = 18	x = 2; y = 2
21	6x+2y = 54; 6x+7y = 99	x = 6; y = 9
22	5x-8y = -107; x+y = 15	x = 1; y = 14
23	x+8y = 60; 8x+y = 39	x = 4; y = 7
24	9x+5y = 68; 8x+3y = 46	x = 2; y = 10
25	3x+y = 25; 5x+8y = 10	x = 10; y = -5
26	6x+3y = -9; x+7y = -47	x = 2; y = -7
27	5x+2y = 57; 5x+3y = 68	x = 7; y = 11
28	2x+7y = 36; 4x+y = 20	x = 4; y = 4
29	2x+y = 24; 8x+4y = 96	x = 8; y = 8
30	8x+8y = 176; 5x+y = 70	x = 12; y = 10
31	9x+4y = 10; 6x+5y = -19	x = 6; y = -11
32	3x-3y = 33; 5x+6y = 88	x = 14; y = 3
33	8x+3y = 126; 9x+5y = 158	x = 12; y = 10
34	2x+6y = 48; 4x+y = 30	x = 6; y = 6
35	9x+6y = 195; 8x+8y = 208	x = 13; y = 13
36	3x+8y = 96; 9x+6y = 126	x = 8; y = 9
37	7x+6y = 49; 7x+4y = 35	x = 1; y = 7
38	2x+7y = 25; 6x+8y = 36	x = 2; y = 3
39	9x+9y = 162; 3x+9y = 114	x = 8; y = 10
40	2x+3y = -19; 7x+9y = -50	x = 7; y = -11
41	7x+2y = 33; x+6y = 39	x = 3; y = 6
42	7x+5y = 129; x+8y = 84	x = 12; y = 9
43	5x+8y = 153; 6x+6y = 144	x = 13; y = 11
44	2x+9y = 91; 6x+6y = 126	x = 14; y = 7
45	x+8y = 106; 5x+2y = 36	x = 2; y = 13
46	x+4y = 25; 9x+6y = 75	x = 5; y = 5
47	5x+8y = 174; 6x+5y = 149	x = 14; y = 13
48	7x+3y = -29; 8x+y = -38	x = -5; y = 2
49	9x+y = 63; 9x+9y = 135	x = 6; y = 9
50	3x+5y = 81; 8x+9y = 164	x = 7; y = 12

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(-3,6) P2(4,8); A(8,8) B(1,6)	2x-7y+48=0, 2x-7y+40=0	parallel lines: m1=2/7 m2=-2/-7
2	P1(0,6) P2(7,9); A(0,8) B(3,1)	3x-7y+42=0, 7x+3y-24=0	perpendicular lines: m1=3/7 m2=-7/3
3	P1(3,5) P2(8,1); A(4,2) B(8,7)	4x+5y-37=0, 5x-4y-12=0	perpendicular lines: m1=-4/5 m2=5/4
4	P1(-6,8) P2(2,2); A(2,7) B(10,1)	3x+4y-14=0, 3x+4y-34=0	parallel lines: m1=-3/4 m2=-3/4
5	P1(-3,7) P2(2,1); A(1,6) B(6,0)	6x+5y-17=0, 6x+5y-36=0	parallel lines: m1=-6/5 m2=-6/5
6	P1(7,6) P2(3,0); A(8,3) B(2,7)	3x-2y-9=0, 2x+3y-25=0	perpendicular lines: m1=-3/-2 m2=2/-3
7	P1(-8,1) P2(2,0); A(3,5) B(13,4)	x+10y-2=0, x+10y-53=0	parallel lines: m1=-1/10 m2=-1/10
8	P1(5,8) P2(6,2); A(0,4) B(6,5)	6x+y-38=0, x-6y+24=0	perpendicular lines: m1=-6/1 m2=1/6
9	P1(-9,7) P2(9,4); A(4,10) B(10,9)	x+6y-33=0, x+6y-64=0	parallel lines: m1=-1/6 m2=-1/6
10	P1(0,0) P2(5,3); A(5,5) B(8,0)	3x-5y-0=0, 5x+3y-40=0	perpendicular lines: m1=3/5 m2=-5/3
11	P1(-7,1) P2(1,2); A(13,3) B(5,2)	x-8y+15=0, x-8y+11=0	parallel lines: m1=1/8 m2=-1/-8
12	P1(9,2) P2(1,0); A(5,9) B(7,1)	x-4y-1=0, 4x+y-29=0	perpendicular lines: m1=-1/-4 m2=-4/1
13	P1(-3,7) P2(7,0); A(10,7) B(0,14)	7x+10y-49=0, 7x+10y-140=0	parallel lines: m1=-7/10 m2=7/-10
14	P1(5,8) P2(3,3); A(8,4) B(3,6)	5x-2y-9=0, 2x+5y-36=0	perpendicular lines: m1=-5/-2 m2=2/-5
15	P1(-4,4) P2(8,8); A(3,2) B(6,3)	x-3y+16=0, x-3y+3=0	parallel lines: m1=1/3 m2=1/3
16	P1(4,9) P2(3,0); A(0,8) B(9,7)	9x-y-27=0, x+9y-72=0	perpendicular lines: m1=-9/-1 m2=-1/9
17	P1(-7,6) P2(5,2); A(2,7) B(11,4)	x+3y-11=0, x+3y-23=0	parallel lines: m1=-1/3 m2=-1/3
18	P1(4,4) P2(6,2); A(5,5) B(7,7)	x+y-8=0, x-y-0=0	perpendicular lines: m1=-1/1 m2=1/1
19	P1(2,6) P2(0,3); A(2,2) B(5,0)	3x-2y+6=0, 2x+3y-10=0	perpendicular lines: m1=-3/-2 m2=-2/3
20	P1(-1,5) P2(2,4); A(7,13) B(10,12)	x+3y-14=0, x+3y-46=0	parallel lines: m1=-1/3 m2=-1/3

Analytic Geometry: Equation of a Line

Given two points, the equation of a line whose points are equidistant from the two given points is determined.

Problem	Given Points	Equation of the Line
1	P1(-5,2) P2(6,-1)	11x-3y-4=0
2	P1(-1,7) P2(-9,-4)	16x+22y+47=0
3	P1(6,9) P2(8,-2)	4x-22y+49=0
4	P1(1,-3) P2(-4,-6)	5x+3y+21=0
5	P1(6,3) P2(-4,2)	20x+2y-25=0
6	P1(5,9) P2(-2,-1)	14x+20y-101=0
7	P1(5,1) P2(5,-2)	0x-2y-1=0
8	P1(-2,5) P2(3,-5)	2x-4y-1=0
9	P1(4,4) P2(-5,5)	9x-1y+9=0
10	P1(9,-3) P2(-3,-4)	24x+2y-65=0
11	P1(1,8) P2(-9,-2)	1x+1y+1=0
12	P1(-5,7) P2(-1,-1)	1x-2y+9=0
13	P1(-4,1) P2(7,-7)	22x-16y-81=0
14	P1(7,6) P2(-5,1)	24x+10y-59=0
15	P1(1,-1) P2(-3,-4)	8x+6y+23=0
16	P1(6,7) P2(-2,-2)	16x+18y-77=0
17	P1(9,7) P2(1,-8)	16x+30y-65=0
18	P1(-1,7) P2(-7,-4)	12x+22y+15=0
19	P1(-3,3) P2(-2,-5)	2x-16y-11=0
20	P1(6,9) P2(3,-6)	1x+5y-12=0
21	P1(9,-9) P2(-1,-3)	5x-3y-38=0
22	P1(1,7) P2(-7,-7)	4x+7y+12=0
23	P1(5,8) P2(-2,4)	14x+8y-69=0
24	P1(-9,4) P2(3,-4)	3x-2y+9=0
25	P1(4,9) P2(5,-1)	2x-20y+71=0
26	P1(5,-2) P2(-9,-7)	28x+10y+101=0
27	P1(9,9) P2(-3,-5)	6x+7y-32=0
28	P1(3,3) P2(-5,6)	16x-6y+43=0
29	P1(-1,1) P2(-2,-7)	2x+16y+51=0
30	P1(-5,8) P2(2,-5)	7x-13y+30=0

Analytic Geometry: Distance Between Two Points

Given two points, their distance and the slope of the line joining them are solved.

Problem	Given Points	Distance	Slope
1	P1(3,7) P2(5,1)	sqrt of 40	-3/1
2	P1(3,2) P2(-8,3)	sqrt of 122	1/-11
3	P1(-3,4) P2(-8,-6)	sqrt of 125	-2/-1
4	P1(7,2) P2(-2,-6)	sqrt of 145	-8/-9
5	P1(5,-9) P2(-8,-1)	sqrt of 233	8/-13
6	P1(6,8) P2(2,4)	sqrt of 32	-1/-1
7	P1(3,6) P2(9,3)	sqrt of 45	-1/2
8	P1(5,6) P2(8,-3)	sqrt of 90	-3/1
9	P1(-5,3) P2(3,-4)	sqrt of 113	-7/8
10	P1(7,2) P2(1,8)	sqrt of 72	1/-1
11	P1(4,1) P2(-6,6)	sqrt of 125	1/-2
12	P1(-8,4) P2(7,-7)	sqrt of 346	-11/15
13	P1(1,3) P2(-7,-1)	sqrt of 80	-1/-2
14	P1(6,9) P2(4,-2)	sqrt of 125	-11/-2
15	P1(8,-7) P2(-7,-9)	sqrt of 229	-2/-15
16	P1(-9,4) P2(-8,-7)	sqrt of 122	-11/1
17	P1(3,2) P2(6,9)	sqrt of 58	7/3
18	P1(5,2) P2(8,5)	sqrt of 18	1/1
19	P1(5,9) P2(9,2)	sqrt of 65	-7/4
20	P1(9,3) P2(7,9)	sqrt of 40	3/-1
21	P1(4,2) P2(-9,-9)	sqrt of 290	-11/-13
22	P1(4,2) P2(9,-8)	sqrt of 125	-2/1
23	P1(1,6) P2(-8,8)	sqrt of 85	2/-9
24	P1(7,6) P2(1,4)	sqrt of 40	-1/-3
25	P1(-9,8) P2(1,-9)	sqrt of 389	-17/10
26	P1(7,8) P2(3,1)	sqrt of 65	-7/-4
27	P1(-9,6) P2(-1,-8)	sqrt of 260	-7/4
28	P1(6,1) P2(9,9)	sqrt of 73	8/3
29	P1(9,9) P2(5,7)	sqrt of 20	-1/-2
30	P1(3,-1) P2(-9,-8)	sqrt of 193	-7/-12
31	P1(4,8) P2(7,1)	sqrt of 58	-7/3
32	P1(9,-9) P2(-8,-2)	sqrt of 338	7/-17
33	P1(2,8) P2(1,4)	sqrt of 17	-4/-1
34	P1(3,2) P2(9,5)	sqrt of 45	1/2
35	P1(3,2) P2(-8,-8)	sqrt of 221	-10/-11
36	P1(9,8) P2(6,-1)	sqrt of 90	-3/-1
37	P1(-5,9) P2(-9,-7)	sqrt of 272	-4/-1
38	P1(5,8) P2(-6,5)	sqrt of 130	-3/-11
39	P1(-8,5) P2(5,-3)	sqrt of 233	-8/13
40	P1(2,9) P2(1,5)	sqrt of 17	-4/-1
41	P1(8,7) P2(6,-5)	sqrt of 148	-6/-1
42	P1(2,8) P2(1,7)	sqrt of 2	-1/-1
43	P1(8,9) P2(4,2)	sqrt of 65	-7/-4
44	P1(4,6) P2(9,3)	sqrt of 34	-3/5
45	P1(-4,2) P2(9,-6)	sqrt of 233	-8/13
46	P1(3,2) P2(4,5)	sqrt of 10	3/1
47	P1(1,4) P2(-7,-7)	sqrt of 185	-11/-8
48	P1(5,3) P2(-9,6)	sqrt of 205	3/-14
49	P1(6,-2) P2(-4,-6)	sqrt of 116	-2/-5
50	P1(-9,2) P2(-1,-4)	sqrt of 100	-3/4

Arithmetic: Division of Decimal Numbers

Problem	Dividend	Divisor	Quotient
1	44.784	56.7892	0.7886
2	44.5097	67.5208	0.6592
3	7888.9234	892.604	8.8381
4	65098.6465	692.8491	93.9579
5	786.7735	2.1861	359.8982
6	64.873	924.1173	0.0702
7	6.6735	7.7284	0.8635
8	59.3669	8.0468	7.3777
9	430.599	76.7433	5.6109
10	0.0016	0.0621	0.0264
11	20.4644	33.6585	0.608
12	0.0015	0.8061	0.0019
13	436.7363	798.5671	0.5469
14	23503.1479	436.0073	53.9054
15	3279.8532	8.5102	385.4026
16	0.156	21.9669	0.0071
17	0.2264	5.7467	0.0394
18	197.5916	76.9318	2.5684
19	26.3738	6.5734	4.0122
20	0.0628	62.7709	0.001
21	6675.0322	85.6386	77.9442
22	181824.5404	631.6969	287.8351
23	0	0.0009	0.0169
24	3.5173	0.0854	41.1864
25	688.9783	0.8178	842.4777
26	6214.5181	69.0601	89.9871
27	0.0195	0.8937	0.0218
28	20.6037	35.6403	0.5781
29	27839.2215	44.1776	630.166
30	562.9067	0.8819	638.2886
31	9035.2309	93.6974	96.4299
32	4389.1909	51.2218	85.6899
33	0.0109	0.0763	0.1431
34	104.512	144.1147	0.7252
35	0.2406	4.2352	0.0568
36	81.2819	12.7477	6.3762
37	7573.4771	181.2986	41.7735
38	3705.9026	453.9767	8.1632
39	29.7328	0.1367	217.5041
40	10.2486	14.0469	0.7296
41	965.5841	116.2934	8.303
42	31.0689	502.7334	0.0618
43	1951.7555	272.0558	7.1741
44	379.0092	7.2848	52.0274
45	20.5656	2.6399	7.7903
46	42017.6013	575.0369	73.0694
47	15.3743	174.5096	0.0881
48	2089.4177	424.3334	4.924
49	197.343	1.3559	145.5439
50	0.0182	0.0653	0.2785