[substitution method][section 28]

(1.)  x = y + 4

2x - 5y = 2               here is the problem

2(y + 4) - 5y = 2        replace x with y + 4

2y + 8 - 5y = 2         multiply thru parentheses

-3y + 8 = 2      combine like terms

3y - 8 = -2       multiply thru by -1

+8  +8   add 8 to each side
________________
_____      ___
3        3        divide each side by 3

y = 2             divide and cancel

x = 2 + 4              replace y with 2

result:  x = 6 and y = 2

(2.)   y = 3 - x

5x + 3y = -1                 here is the problem

5x + 3(3 - x) = -1    replace y with 3 - x

5x + 9 - 3x = -1        multiply thru parentheses

2x + 9 = -1           combine like terms

-9   -9           subtract 9 from each side
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2x    =  -10         subtract
____     _______
2       2             divide each side by 2

x = -5           divide and cancel

y = 3 - (-5)     replace x with -5

result: x = -5 and y = 8

(3.)  x = 8y

x - 4y = 12              here is the problem

8y - 4y = 12         replace x with 8y

4y = 12           combine like terms
____ ____
4     4       divide each side by 4

y = 3           divide and cancel

x = 8(3)              replace y with 3

x = 24          multiply

result:   x = 24 and y = 3

(4.)  5x - y = 1

3x + 2y = 11               here is the problem

-5x + y = -1      multiply equation 1 thru by -1

+ 5x    + 5x      add 5x to each side
________________
y = 5x - 1       add

3x + 2(5x - 1) = 11   replace y with 5x - 1

3x + 10x - 2 = 11      multiply thru parentheses

13x - 2 = 11         combine like terms

+ 2  + 2     add 2 to each side
___________________
____         ___
13            13     divide each side by 13

x = 1         divide and cancel

y = 5(1) - 1     replace x with 1

y = 4         multiply & combine like terms

result:  x = 1 and y = 4

(5.)   5a = b

a = 10b - 49           here is the problem

a = 10(5a) - 49        replace b with 5a

a = 50a - 49         multiply

-a = -50a + 49      multiply thru by -1

+50a  +50a          add 50a to each side
____________________
___         ____
49          49       divide each side by 49

a = 1            divide and cancel

b = 5(1)           replace a with 1

b = 5              multiply

result:  a = 1 and b = 5

(6.)  r = 5 - s

2r + 7s = 20                 here is the problem

2(5 - s) + 7s = 20      replace r with 5 - s

10 - 2s + 7s = 20        multiply thru parentheses

10 + 5s = 20            combine like terms

-10     -10      subtract 10 from each side
______________________
5s = 10        subtract
___  ___
5     5        divide each side by 5

s = 2            divide and cancel

r = 5 - 2      replace s with 2

r = 3             subtract

result:  r = 3 and s = 2

(7.)  4x - y = 15

5x - 8y = -15                   here is the problem

-4x + y = -15          multiply equation 1 thru by -1

+4x    + 4x       add 4x to each side
_________________
y = 4x - 15        add

5x - 8(4x - 15) = -15    replace y with 4x - 15

5x - 32x + 120 = -15      multiply thru parentheses

-27x + 120 = -15           combine like terms

27x - 120 = 15      multiply thru by -1

+120  +120   add 120 to each side
____________________
______      ____
27           27        divide each side by 27

x = 5             divide and cancel

y  = 4(5) - 15          replace x with 5

y = 5                multiply and combine like terms

result: x = 5 and y = 5

(8.)   r + s = 3

r - s = 1                 here is the problem

+  s  + s           add s to each side
____________
r   = s + 1         add

s + 1 + s = 3            replace r with s + 1

2s + 1 = 3             combine like terms

-1  -1     subtract 1 from each side
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2s   =   2        subtract
____    ____
2         2           divide each side by 2

s = 1            divide and cancel

r = 1 + 1             replace s  with 1

result:  r = 2 and s = 1

(9.)  p - 3q = 0

2p - 5q = 2                      here is the problem

p - 3q = 0

+  3q  +3q       add 3q to each side
__________________

2(3q) - 5q = 2   replace p with 3q

6q - 5q = 2             multiply

q = 2                   combine like terms

p = 3(2)             replace q with 2

p = 6              multiply

result:  p = 6 and q = 2

(10.)  2a + b = 5

8a - b = -5                  here is the problem

-8a + b = 5   multiply equation 2 thru by -1

+8a    +8a    add 8a to each side
_________________
b = 8a + 5       add

2a + 8a + 5 = 5      replace b with 8a + 5

10a + 5 = 5         combine like terms

-   5  -5      subtract 5 from each side
______________________
10a         = 0     subtract
____         ____
10            10        divide each side by 10

a = 0                divide and cancel

b = 8(0) + 5    replace a with 0

b = 5              multiply & add

result:  a = 0 and b = 5