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2/3n - 2/3 = n/6 + 4/3

 Apr 20, 2016
 #1
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Solve for n:             (2/3)n - 2/3 = n/6 + 4/3. I have taken the 1st. term as shown
(2 n)/3-2/3 = n/6+4/3

 

(2 n)/3-2/3 = (2 n-2)/3:
(2 n-2)/3 = n/6+4/3

 

Put each term in n/6+4/3 over the common denominator 6: n/6+4/3  =  n/6+8/6:
(2 n-2)/3 = n/6+8/6

 

n/6+8/6 = (n+8)/6:
(2 n-2)/3 = (n+8)/6

 

Multiply both sides by 6:
(6 (2 n-2))/3 = (6 (n+8))/6

 

6/3 = (3×2)/3 = 2:
2 (2 n-2) = (6 (n+8))/6

 

(6 (n+8))/6 = 6/6×(n+8) = n+8:
2 (2 n-2) = n+8

 

Expand out terms of the left hand side:
4 n-4 = n+8

 

Subtract n from both sides:
(4 n-n)-4 = (n-n)+8

 

4 n-n = 3 n:
3 n-4 = (n-n)+8

 

n-n = 0:
3 n-4 = 8

 

Add 4 to both sides:
3 n+(4-4) = 8+4

 

4-4 = 0:
3 n = 8+4

 

8+4 = 12:
3 n = 12

 

Divide both sides of 3 n = 12 by 3:
(3 n)/3 = 12/3

 

3/3 = 1:
n = 12/3

 

The gcd of 12 and 3 is 3, so 12/3 = (3×4)/(3×1) = 3/3×4 = 4:
Answer: |  n = 4

 Apr 21, 2016
 #2
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\(\frac{2}{3n}-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}\)

 

\(\frac{6}{9n}-\frac{6n}{9n}=\frac{n}{6}+\frac{4}{3}\)

 

\(\frac{6-6n}{9n}=\frac{n}{6}+\frac{4}{3}\)

 

\(\frac{6-6n}{9n}=\frac{2n}{12}+\frac{16}{12}\)

 

\(\frac{6-6n}{9n}=\frac{2n+16}{12}\)

 

\(6-6n=\frac{9n\times(2n+16)}{12}\)

 

\(6-6n=\frac{3n\times(2n+16)}{4}\)

 

\(4\times(6-6n)=3n\times(2n+16)\)

 

\(24-24n=3n\times(2n+16)\)

 

\(24-24n={6n}^{2}+48n\)

 

\(24={6n}^{2}+72n\)

 

\(4={n}^{2}+12n\)

 

\(40={n}^{2}+12n+36\)

 

\(40={(n+6)}^{2}\)

 

\(±2\sqrt{10}=n+6\)

 

\(-6±2\sqrt{10}=n\)

 

\(n=-6±2\sqrt{10}\)

 

\(n=-6+2\sqrt{10};\)  \(n≈ 0.3245553203367587\)

 

\(n=-6-2\sqrt{10};\)  \(n≈-12.3245553203367587\)

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 Apr 21, 2016
 #3
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n/6+3=12

 Apr 26, 2016

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