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A number besides 0. That both multiplied by 0.5 and added to 0.5 gives the same result

Guest Feb 26, 2017
 #1
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A number besides 0. That both multiplied by 0.5 and added to 0.5 gives the same result

 

a x 1/2 =b

a + 1/2 =b

 

Solve the following system:
{a/2 = b | (equation 1)
a + 1/2 = b | (equation 2)
Express the system in standard form:
{a/2 - b = 0 | (equation 1)
a - b = -1/2 | (equation 2)
Swap equation 1 with equation 2:
{a - b = (-1)/2 | (equation 1)
a/2 - b = 0 | (equation 2)
Subtract 1/2 × (equation 1) from equation 2:
{a - b = -1/2 | (equation 1)
0 a - b/2 = 1/4 | (equation 2)
Multiply equation 1 by 2:
{2 a - 2 b = -1 | (equation 1)
0 a - b/2 = 1/4 | (equation 2)
Multiply equation 2 by 4:
{2 a - 2 b = -1 | (equation 1)
0 a - 2 b = 1 | (equation 2)
Divide equation 2 by -2:
{2 a - 2 b = -1 | (equation 1)
0 a+b = (-1)/2 | (equation 2)
Add 2 × (equation 2) to equation 1:
{2 a+0 b = -2 | (equation 1)
0 a+b = -1/2 | (equation 2)
Divide equation 1 by 2:
{a+0 b = -1 | (equation 1)
0 a+b = -1/2 | (equation 2)
Collect results:
Answer: |a = -1            and             b= -1/2

Guest Feb 26, 2017
 #2
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a x 1/2 =b..........(1)

a + 1/2 =b..........(2)

 

From 2 above we have:

a = b - 1/2, we sub into (1)

(b -1/2) x 1/2 =b

1/2b - 1/4 = b

- 1/4 = b - 1/2b

- 1/4 = 1/2b divide both sides by 1/2

b = - 1/4 x 2

b = - 1/2 sub into (2) above and we have:

a + 1/2 = - 1/2

a = -1/2 - 1/2

a = - 1

Guest Feb 26, 2017

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