3 variable linear equation?

Question

For the life of my I cannot solve this equation I know how, but the numbers are so hard to change them to get same coefficients that I always end with with x = 2.34523 and I am fairly sure that the variables equal whole numbers because this is a word problem about price per item.

equation:

1b + 100m + 16d = 435
3b + 18 m + 45d= 435
7b + 6 m + 46d= 435

Thanks for the help...

Answer 1

let b= 435-100m-16d

3(435-100m-16d) + 18m + 45d = 435
1305 - 300m - 48d +18m +45d = 435
282m+3d=870 ----(1)

7(435-100m-16d) + 6m + 46d = 435
3045-700m-112d +6m + 46d = 435
694m + 66d = 2610 -----(2)

(1) * 22

6204m + 66d = 19140

(1) - (2)
6204m-694m +66d - 66d = 19140-2610
5510m=16530
m=3

sub m=3 into (1)
282(3)+3d=870
3d=870 - 846
d=24/3
d= 8

b+100(3) +16(8) = 435
b=435-300-128
b=7

ans = m=3, d=8, b=7

Answer 2

Finding b:
b + 100m + 16d = 435
b = 435 - 100m - 16d

Finding d:
3b + 18 m + 45d= 435
Plug b with 435 - 100m - 16d
3(435 - 100m - 16d) + 18m + 45d = 435
1,305 - 300m - 48d + 18m + 45d = 435
- 3d = - 870 + 282m
d = 290 - 94m

b = 435 - 100m - 16d
Finding b (plug d with 290 - 94m)
b = 435 - 100m - 16(290 - 94m)
b = 435 - 100m - 4,640 + 1,504m
b = 1,404m - 4,205

7b + 6m + 46d= 435
Finding m (plug b with 1,404m - 4,205 and d with 290 - 94m):
7(1,404m - 4,205) + 6m + 46(290 - 94m) = 435
9,828m - 29,435 + 6m + 13,340 - 4,324m = 435
5,510m = 16,530
m = 3

Finding b (plug m with 3):
b = 1,404m - 4,205
b = 1,404(3) - 4,205
b = 7

Finding d (plug m with 3):
d = 290 - 94m
d = 290 - 94(3)
d = 290 - 282
d = 8

Answer: b = 7, d = 8, m = 3

Proofs (total is 435 in all the above original equations):
7 + 100(3) + 16(8) = 3(7) + 18(3) + 45(8) = 7(7) + 6(3) + 46(8)
7 + 300 + 128 = 21 + 54 + 360 = 49 + 18 + 368
435 = 435 = 435

Answer 3

You need to combine two equations together and then another two together. This way, you get rid of one of the variables.
Combine equations (1) and (2).
Multiply (1) by 3 to eliminate the b from (2): b(1b+ 100m + 16d) = 3(435)
3b + 300m + 48d = 1305

Subtract (1) and (2)
3b + 300m + 48d = 1305
3b + 18m + 45d = 435

282m + 3d = 870 First combined equation

Combine (1) and (3)
Multiply (1) by 7: 7(1b + 100m + 16d = 435)
7b + 700m + 112d = 3045

Subtract (1) and (3)
7b + 700m + 112d = 3045
7b + 6m + 46d = 435

694m + 66d = 2610 2nd combined equation

694 m + 66d = 2610
22(282m + 3d = 870)

Subtract
694m + 66d = 2610
6204m + 66d = 19140

-5510m = -16530
m = 3

Find d: 282m + 3d = 870
282(3) + 3d = 870
3d = 24
d = 8

Find b: 3b + 18m + 45d = 435
3b + 18(3) + 45(8) = 435
3b = 21
b = 7

Answer 4

1b + 100m + 16d = 435
3b + 300m + 48d = 1305
21b + 2100m + 336d = 9135
21b = 9135 - 2100m - 336d

1b + 100m + 16d = 3b + 18 m + 45d = 7b + 6 m + 46d
100m + 16d =18 m + 45d = 3b + 6 m + 46d
76m + 16d = 45d = 3b + 46d
76m = 45d -16d - 46d = 3b
76m = -17d = 3b
76m = 3b + 17d


3b + 18 m + 45d= 435
7b + 6 m + 46d= 435

21b = 9135 - 2100m - 336d
21b = 3045 - 126m - 315d
0 = 6090 - 1974m -21d
1974m + 21d = 6090
658m + 7d = 2030

A GOOD START

7b + 6 m + 46d= 435
3b + 18 m + 45d= 435
4b - 12 m + d = 0
4b + d = 12m
d = 12m - 4b

21b = 1305 - 18m + 138d

b = 435 - 100m -16d

Answer 5

{b = 7, m = 3, d = 8}

Set it up in a matrix and use Gauss Jordan elimination or Kramer's rule. Thoes are methods are too hard to type into this box, sorry.

Answer 6

b=7
m=3
d=8