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
+1904 \(\frac{2}{3n}-\frac{2}{3}=\frac{n}{6}+\frac{4}{3}\)
\(\frac{6}{9n}-\frac{6n}{9n}=\frac{n}{6}+\frac{4}{3}\)
\(\frac{6-6n}{9n}=\frac{n}{6}+\frac{4}{3}\)
\(\frac{6-6n}{9n}=\frac{2n}{12}+\frac{16}{12}\)
\(\frac{6-6n}{9n}=\frac{2n+16}{12}\)
\(6-6n=\frac{9n\times(2n+16)}{12}\)
\(6-6n=\frac{3n\times(2n+16)}{4}\)
\(4\times(6-6n)=3n\times(2n+16)\)
\(24-24n=3n\times(2n+16)\)
\(24-24n={6n}^{2}+48n\)
\(24={6n}^{2}+72n\)
\(4={n}^{2}+12n\)
\(40={n}^{2}+12n+36\)
\(40={(n+6)}^{2}\)
\(±2\sqrt{10}=n+6\)
\(-6±2\sqrt{10}=n\)
\(n=-6±2\sqrt{10}\)
\(n=-6+2\sqrt{10};\) \(n≈ 0.3245553203367587\)
\(n=-6-2\sqrt{10};\) \(n≈-12.3245553203367587\)
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