Solve 1/2(2x+3)+1/3(2x+4)=1/4(2x+5)+1/5(2x+6)

$$\frac{ 1 }{ 2 } (2x+3)+ \frac{ 1 }{ 3 } (2x+4)= \frac{ 1 }{ 4 } (2x+5)+ \frac{ 1 }{ 5 } (2x+6)$$

Solve for x

$$x=-\frac{1}{2}=-0.5$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2} by 2x+3.

$$\frac{1}{2}\times 2x+\frac{1}{2}\times 3+\frac{1}{3}\left(2x+4\right)=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Cancel out 2 and 2.

$$x+\frac{1}{2}\times 3+\frac{1}{3}\left(2x+4\right)=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$x+\frac{3}{2}+\frac{1}{3}\left(2x+4\right)=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Use the distributive property to multiply \frac{1}{3} by 2x+4.

$$x+\frac{3}{2}+\frac{1}{3}\times 2x+\frac{1}{3}\times 4=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Multiply \frac{1}{3} and 2 to get \frac{2}{3}.

$$x+\frac{3}{2}+\frac{2}{3}x+\frac{1}{3}\times 4=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Multiply \frac{1}{3} and 4 to get \frac{4}{3}.

$$x+\frac{3}{2}+\frac{2}{3}x+\frac{4}{3}=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Combine x and \frac{2}{3}x to get \frac{5}{3}x.

$$\frac{5}{3}x+\frac{3}{2}+\frac{4}{3}=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Least common multiple of 2 and 3 is 6. Convert \frac{3}{2} and \frac{4}{3} to fractions with denominator 6.

$$\frac{5}{3}x+\frac{9}{6}+\frac{8}{6}=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Since \frac{9}{6} and \frac{8}{6} have the same denominator, add them by adding their numerators.

$$\frac{5}{3}x+\frac{9+8}{6}=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Add 9 and 8 to get 17.

$$\frac{5}{3}x+\frac{17}{6}=\frac{1}{4}\left(2x+5\right)+\frac{1}{5}\left(2x+6\right)$$

Use the distributive property to multiply \frac{1}{4} by 2x+5.

$$\frac{5}{3}x+\frac{17}{6}=\frac{1}{4}\times 2x+\frac{1}{4}\times 5+\frac{1}{5}\left(2x+6\right)$$

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

$$\frac{5}{3}x+\frac{17}{6}=\frac{2}{4}x+\frac{1}{4}\times 5+\frac{1}{5}\left(2x+6\right)$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{5}{3}x+\frac{17}{6}=\frac{1}{2}x+\frac{1}{4}\times 5+\frac{1}{5}\left(2x+6\right)$$

Multiply \frac{1}{4} and 5 to get \frac{5}{4}.

$$\frac{5}{3}x+\frac{17}{6}=\frac{1}{2}x+\frac{5}{4}+\frac{1}{5}\left(2x+6\right)$$

Use the distributive property to multiply \frac{1}{5} by 2x+6.

$$\frac{5}{3}x+\frac{17}{6}=\frac{1}{2}x+\frac{5}{4}+\frac{1}{5}\times 2x+\frac{1}{5}\times 6$$

Multiply \frac{1}{5} and 2 to get \frac{2}{5}.

$$\frac{5}{3}x+\frac{17}{6}=\frac{1}{2}x+\frac{5}{4}+\frac{2}{5}x+\frac{1}{5}\times 6$$

Multiply \frac{1}{5} and 6 to get \frac{6}{5}.

$$\frac{5}{3}x+\frac{17}{6}=\frac{1}{2}x+\frac{5}{4}+\frac{2}{5}x+\frac{6}{5}$$

Combine \frac{1}{2}x and \frac{2}{5}x to get \frac{9}{10}x.

$$\frac{5}{3}x+\frac{17}{6}=\frac{9}{10}x+\frac{5}{4}+\frac{6}{5}$$

Least common multiple of 4 and 5 is 20. Convert \frac{5}{4} and \frac{6}{5} to fractions with denominator 20.

$$\frac{5}{3}x+\frac{17}{6}=\frac{9}{10}x+\frac{25}{20}+\frac{24}{20}$$

Since \frac{25}{20} and \frac{24}{20} have the same denominator, add them by adding their numerators.

$$\frac{5}{3}x+\frac{17}{6}=\frac{9}{10}x+\frac{25+24}{20}$$

Add 25 and 24 to get 49.

$$\frac{5}{3}x+\frac{17}{6}=\frac{9}{10}x+\frac{49}{20}$$

Subtract \frac{9}{10}x from both sides.

$$\frac{5}{3}x+\frac{17}{6}-\frac{9}{10}x=\frac{49}{20}$$

Combine \frac{5}{3}x and -\frac{9}{10}x to get \frac{23}{30}x.

$$\frac{23}{30}x+\frac{17}{6}=\frac{49}{20}$$

Subtract \frac{17}{6} from both sides.

$$\frac{23}{30}x=\frac{49}{20}-\frac{17}{6}$$

Least common multiple of 20 and 6 is 60. Convert \frac{49}{20} and \frac{17}{6} to fractions with denominator 60.

$$\frac{23}{30}x=\frac{147}{60}-\frac{170}{60}$$

Since \frac{147}{60} and \frac{170}{60} have the same denominator, subtract them by subtracting their numerators.

$$\frac{23}{30}x=\frac{147-170}{60}$$

Subtract 170 from 147 to get -23.

$$\frac{23}{30}x=-\frac{23}{60}$$

Multiply both sides by \frac{30}{23}, the reciprocal of \frac{23}{30}.

$$x=-\frac{23}{60}\times \frac{30}{23}$$

Multiply -\frac{23}{60} times \frac{30}{23} by multiplying numerator times numerator and denominator times denominator.

$$x=\frac{-23\times 30}{60\times 23}$$

Do the multiplications in the fraction \frac{-23\times 30}{60\times 23}.

$$x=\frac{-690}{1380}$$

Reduce the fraction \frac{-690}{1380} to lowest terms by extracting and canceling out 690.

$$x=-\frac{1}{2}$$