[elimination method][section 26]

(1.)  3x + y = 9                here is the problem

2x + y = 1

3x + y = 9             put this here

-2x - y = -1         multiply equation 1 thru by -1
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x   =    8           subtract equations

3(8) + y = 9        replace x with 8

24 + y = 9         multiply

-24     -24     subtract 24 from each side
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y = -15     subtract

result:   x = 8 and y = -15

(2.)  5r - s = -23

3r - s = -15                     here is the problem

5r - s = -23            put this here

-3r + s = 15        multiply equation 2 thru by -1
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2r     = -8           add equations
____     ____
2         2        divide each side by 2

r = -4            divide and cancel

3(-4) - s = -15       replace r with -4

-12 - s = -15           multiply

12 + s = 15          multiply thru by -1

-12          -12   subtract 12 from each side
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s = 3         subtract

result:  r = -4  and s = 3

(3.)   2x + 3y = 8

3x + y = 5                  here is the problem

9x + 3y = 15         multiply equation 2 thru by 3

-2x - 3y = -8        multiply equation 1 thru by -1
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7x      =   7           subtract equations
____        ____
7          7          divide each side by 7

x = 1               divide and cancel

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

3 + y = 5               multiply

-3     -3         subtract 3 from each side
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y = 2          subtract

result:   x = 1  and y = 2

(4.)  2x + y = 3

7x - 4y = 18                       here is the problem

8x + 4y = 12           multiply equation 1 thru by 4
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15x   =   30       add equations
_____     ____
15         15         divide each side by 15

x = 2           divide and cancel

2(2) + y = 3          replace x with 2

4 + y = 3             multiply

-4       -4            subtract 4 from each side
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y = -1            subtract

result:  x = 2 and y = -1

(5.)  5x + y = 15

3x + 2y = 9                     here is the problem

10x + 2y = 30         multiply equation 1 thru by 2

-3x - 2y = -9         multiply equation 2 thru by -1
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7x     =  21        subtract equations
___       ___
7         7        divide each side by 7

x = 3         divide and cancel

5(3) + y = 15          replace x with 3

15 + y = 15            multiply

-15      -15         subtract 15 from each side
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y = 0           subtract

result:  x = 3 and y = 0

(6.)    7r - 5t = -2

-8r - t = 9                   here is the problem

40r + 5t = -45        multiply equation 2 thru by -5

7r - 5t = -2        put this here
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47r     = -47          add equations
_____      _____
47         47          divide each side by 47

r = -1              divide and cancel

-8(-1) - t = 9          replace r with -1

8 - t = 9               multiply

-8 + t = -9       multiply thru by -1

+ 8        +8   add 8 to each side
_____________________
t = -1      add

result:  r = -1  and t = -1

(7.)  4p - 2q = 20

p + 5q = -17                    here is the problem

-4p - 20q = 68        multiply equation 2 thru by -4

4p - 2q = 20         put this here
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-22q = 88        add equations
____  ___
-22  -22       divide each side by -22

q = -4           divide and cancel

p + 5(-4) = -17          replace q with -4

p - 20 = -17           multiply

+    20  + 20      add 20 to each side
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p       =  3           add

result:  p = 3  and  q = -4

(8.)   2a + 5b = 18

-5a + b = -18                 here is the problem

25a - 5b = 90      multiply equation 2 thru by -5

2a + 5b = 18     put this here
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27a     = 108       add equations
____     _____
27       27       divide each side by 27

a = 4             divide and cancel

2(4) + 5b = 18    replace a with 4

8 + 5b = 18         multiply

-8       -8       subtract 8 from each side
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5b = 10       subtract
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5    5         divide each side by 5

b = 2             divide and cancel

result:  a = 4 and b = 2

(9.)   2u + 2z = 8

5u - 3z = 4                     here is the problem

6u + 6z = 24       multiply equation 1 thru by 3

10u - 6z = 8        multiply equation 2 thru by 2
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16u    =  32      add equations
____      ___
16        16         divide each side by 16

u = 2               divide and cancel

2(2) + 2z = 8          replace u with 2

4 + 2z = 8            multiply

-4       -4         subtract 4 from each side
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2z = 4            subtract
___ ___
2    2           divide each side by 2

z = 2          divide and cancel

result:  u = 2 and z = 2

(10.)   4a + 3b = -2

8a - 2b = 12               here is the problem

-8a - 6b = 4        multiply equation 1 thru by -2
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-8b = 16        add equations
__  ____
-8   -8          divide each side by -8

b = -2             divide and cancel

4a + 3(-2) = -2     replace b with -2

4a - 6 = -2              multiply

+ 6  +6          add 6 to each side
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4a        =  4           add
___          ___
4           4        divide each side by 4

a = 1              divide and cancel

result:  a = 1 and b = -2