Solve 3x+14left(5x/4-9/7right)=13x/2-6left(23x/21-frac{7}{12}right)

$$3x+14 \left( \frac{ 5x }{ 4 } - \frac{ 9 }{ 7 } \right) = \frac{ 13x }{ 2 } -6 \left( \frac{ 23x }{ 21 } - \frac{ 7 }{ 12 } \right)$$

Solve for x

$$x = \frac{301}{288} = 1\frac{13}{288} \approx 1.045138889$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 84, the least common multiple of 4,7,2,21,12.

$$252x+1176\left(\frac{5x}{4}-\frac{9}{7}\right)=42\times 13x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 7 is 28. Multiply \frac{5x}{4} times \frac{7}{7}. Multiply \frac{9}{7} times \frac{4}{4}.

$$252x+1176\left(\frac{7\times 5x}{28}-\frac{9\times 4}{28}\right)=42\times 13x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

Since \frac{7\times 5x}{28} and \frac{9\times 4}{28} have the same denominator, subtract them by subtracting their numerators.

$$252x+1176\times \frac{7\times 5x-9\times 4}{28}=42\times 13x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

Do the multiplications in 7\times 5x-9\times 4.

$$252x+1176\times \frac{35x-36}{28}=42\times 13x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

Cancel out 28, the greatest common factor in 1176 and 28.

$$252x+42\left(35x-36\right)=42\times 13x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

Use the distributive property to multiply 42 by 35x-36.

$$252x+1470x-1512=42\times 13x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

Combine 252x and 1470x to get 1722x.

$$1722x-1512=42\times 13x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

Multiply 42 and 13 to get 546.

$$1722x-1512=546x-504\left(\frac{23x}{21}-\frac{7}{12}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 21 and 12 is 84. Multiply \frac{23x}{21} times \frac{4}{4}. Multiply \frac{7}{12} times \frac{7}{7}.

$$1722x-1512=546x-504\left(\frac{4\times 23x}{84}-\frac{7\times 7}{84}\right)$$

Since \frac{4\times 23x}{84} and \frac{7\times 7}{84} have the same denominator, subtract them by subtracting their numerators.

$$1722x-1512=546x-504\times \frac{4\times 23x-7\times 7}{84}$$

Do the multiplications in 4\times 23x-7\times 7.

$$1722x-1512=546x-504\times \frac{92x-49}{84}$$

Cancel out 84, the greatest common factor in 504 and 84.

$$1722x-1512=546x-6\left(92x-49\right)$$

Use the distributive property to multiply -6 by 92x-49.

$$1722x-1512=546x-552x+294$$

Combine 546x and -552x to get -6x.

$$1722x-1512=-6x+294$$

Add 6x to both sides.

$$1722x-1512+6x=294$$

Combine 1722x and 6x to get 1728x.

$$1728x-1512=294$$

Add 1512 to both sides.

$$1728x=294+1512$$

Add 294 and 1512 to get 1806.

$$1728x=1806$$

Divide both sides by 1728.

$$x=\frac{1806}{1728}$$

Reduce the fraction \frac{1806}{1728} to lowest terms by extracting and canceling out 6.

$$x=\frac{301}{288}$$