Given a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5,find the value of a + b + c + d.?

Question

Best Answer is given to first answerr that shows the most work and how their answer is true and how do to do the problem. Good luck.

Answer 1

a+1=b+2
a=b+1

by using algebra like that, you get:
a = b+1 = c+2 = d+3

also, you know that:
a+1 = a + b + c + d + 5
a = a+b+c+d+4
0 = b + c + d + 4
so put that in terms of a:
0 = a - 1 + a - 2 + a - 3 + 4
0= 3a -6 +4
0= 3a -2
-3a = -2
a = 2/3

a + b + c + d = answer
2/3 + 2/3 -1 +2/3 - 2 +2/3 -3 = answer
4(2/3) -6 =answer
8/3 -6 = answer
answer = -3.333333333333333= -3 and 1/3

I'm pretty sure that's right

Answer 2

From the given equations, we have the following:

a + 1 = a + b + c + d + 5
   means b + c + d + 4 = 0
   means b + c + d = -4    (1)

b + 2 = a + b + c + d + 5
   means a + c + d + 3 = 0
   means a + c + d = -3    (2)

c + 3 = a + b + c + d + 5
   means a + b + d + 2 = 0
   means a + b + d = -2    (3)

d + 4 = a + b + c + d + 5
   means a + b + c + 1 = 0
   means a + b + c = -1    (4)

Now, we have the following system of linear equations in four(4)
unknowns expressed as follows:

   => b + c + d = -4    (1)
   => a + c + d = -3    (2)
   => a + b + d = -2    (3)
   => a + b + c = -1    (4)


Using an augmented matrix (in the form [A:b]), each row entry corresponds to the equation we have above and we only Write the coefficients of the unknowns a, b, c and d as follows:
/   0    1    1    1   : -4 \
|   1    0    1    1   : -3 |
|   1    1    0    1   : -2 |
\   1    1    1    0   : -1 /.

Note: The first column of the above matrix corresponds to the coefficients of a. Similarly for the second, third and fourth column corresponds to the coefficients of b, c and d, respectively.

By using the Gauss-Jordan reduction method to soluve for the unknowns, we have the following:(our goal is to obtain an augmented matrix of the form [I:b'], where I is an identity matrix)

*by interchanging row 1 (R1) and row 2 (R2):
/   1    0    1    1   : -3 \
|   0    1    1    1   : -4 |
|   1    1    0    1   : -2 |
\   1    1    1    0   : -1 /

*by interchanging row 3 (R3) and row 4 (R4):
/   1    0    1    1   : -3 \
|   0    1    1    1   : -4 |
|   1    1    1    0   : -1 |
\   1    1    0    1   : -2 /

*-R1 + R3; -R1 + R4:
/   1    0    1    1   : -3 \
|   0    1    1    1   : -4 |
|   0    1    0   -1   :   2 |
\   0    1   -1    0   :   1 /

*-R2 + R3; -R2 + R4:
/   1    0    1    1   : -3 \
|   0    1    1    1   : -4 |
|   0    0   -1   -2   :   6 |
\   0    0   -2   -1   :   5 /

*-R3; (-1/2)R4:
/   1    0    1    1   : -3 \
|   0    1    1    1   : -4 |
|   0    0    1    2   : -6 |
\   0    0    1   1/2   : -5/2/

*-R3 + R1; -R3 + R2 ; -R3 + R4:
/   1    0    0  -1   :   3 \
|   0    1    0  -1   :   2 |
|   0    0    1    2   : -6 |
\   0    0    0   -3/2   :  7/2/

*(-2/3)R4:
/   1    0    0  -1   :   3 \
|   0    1    0  -1   :   2 |
|   0    0    1    2   : -6 |
\   0    0    0   1   :  -7/3/

*-R4 + R1; -R4 + R2; -2R4 + R3;
/   1    0    0  0   :   2/3\
|   0    1    0  0   :  -1/3|
|   0    0    1    0   : -4/3|
\   0    0    0   1   :  -7/3/

Observe that we obtain an identity matrix of order 4 from the augmented matrix above. Moreover, the rightmost column is the value of the unknown a, b, c and d, respectively. That is,

a = 2/3,
b = -1/3,
c = -4/3 and
d = -7/3.


Now, if we plug these values on the given equations, we get the following results
a + 1 = 2/3 + 1 = 5/3
b + 2 = -1/3 + 2 = -1/3 + 6/3 = 5/3
c + 3 = -4/3 + 3 = -4/3 + 9/3 = 5/3
d + 4 = -7/3 + 4 = -7/3 + 12/3 = 5/3 and
a + b + c + d + 5 = 2/3 + (-1/3) + (-4/3) + (-7/3) + 5 = -10/3 + 15/3 = 5/3.

Therefore, a + 1 = b + 2 = c + 3 = d + 4 = a + b + c + d + 5 = 5/3.

Answer 3

a+b+c+d+5 = a+1
b+c+d+4 = 0
c+3=d+4
c+3-d-4 = 0
b+c+d+4 = c-d-1
b+2d+5 = 0
b+2=d+4
b+2-d-4 = 0
b+2d+5 =b-d-2
3d = -7
d = -7/3
-7/3 + 4 = 5/3
so a = 2/3
b = -1/3
c = -4/3

a+b+c+d = -10/3

Answer 4

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Answer 5

let say
a + 1 = b + 2 = c + 3 = d + 4 = x
therefore a=x-1, b=x-2...etc
so (x-1)+(x-2)...etc = -5
so 4x - 10 = -5
so x = 5/4
so a = 1/4
b = -3/4
c = -7/4
d = -11/4

Answer 6

a+1=b+2=c+3=d+4=a+b+c+d+5
a=b+1=c+3=d+4=a+b+c+d+5
a=b=c+2=d+4=a+b+c+d+5
a=b=c=d+2=a+b+c+d+5
a=b=c=d=a+b+c+d+3
then a=b=c=d=a+b+c+d=0