Solve 2/5(x-1)(2x-3)+x-1/7-1/5(x-3)(2x-3)=2/5x^2+frac{1}{7}

$$\frac { 2 } { 5 } ( x - 1 ) ( 2 x - 3 ) + \frac { x - 1 } { 7 } - \frac { 1 } { 5 } ( x - 3 ) ( 2 x - 3 ) = \frac { 2 } { 5 } x ^ { 2 } + \frac { 1 } { 7 }$$

Solve for x

$$x = -\frac{31}{2} = -15\frac{1}{2} = -15.5$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 35, the least common multiple of 5,7.

$$35\left(\frac{2}{5}\left(x-1\right)\left(2x-3\right)+\frac{x-1}{7}\right)-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

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

$$35\left(\left(\frac{2}{5}x-\frac{2}{5}\right)\left(2x-3\right)+\frac{x-1}{7}\right)-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Use the distributive property to multiply \frac{2}{5}x-\frac{2}{5} by 2x-3 and combine like terms.

$$35\left(\frac{4}{5}x^{2}-2x+\frac{6}{5}+\frac{x-1}{7}\right)-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5 and 7 is 35. Multiply \frac{6}{5} times \frac{7}{7}. Multiply \frac{x-1}{7} times \frac{5}{5}.

$$35\left(\frac{4}{5}x^{2}-2x+\frac{6\times 7}{35}+\frac{5\left(x-1\right)}{35}\right)-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Since \frac{6\times 7}{35} and \frac{5\left(x-1\right)}{35} have the same denominator, add them by adding their numerators.

$$35\left(\frac{4}{5}x^{2}-2x+\frac{6\times 7+5\left(x-1\right)}{35}\right)-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Do the multiplications in 6\times 7+5\left(x-1\right).

$$35\left(\frac{4}{5}x^{2}-2x+\frac{42+5x-5}{35}\right)-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Combine like terms in 42+5x-5.

$$35\left(\frac{4}{5}x^{2}-2x+\frac{37+5x}{35}\right)-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Use the distributive property to multiply 35 by \frac{4}{5}x^{2}-2x+\frac{37+5x}{35}.

$$28x^{2}-70x+35\times \frac{37+5x}{35}-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Express 35\times \frac{37+5x}{35} as a single fraction.

$$28x^{2}-70x+\frac{35\left(37+5x\right)}{35}-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Cancel out 35 and 35.

$$28x^{2}-70x+37+5x-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Combine -70x and 5x to get -65x.

$$28x^{2}-65x+37-7\left(x-3\right)\left(2x-3\right)=14x^{2}+5$$

Subtract 14x^{2} from both sides.

$$28x^{2}-65x+37-7\left(x-3\right)\left(2x-3\right)-14x^{2}=5$$

Use the distributive property to multiply -7 by x-3.

$$28x^{2}-65x+37+\left(-7x+21\right)\left(2x-3\right)-14x^{2}=5$$

Use the distributive property to multiply -7x+21 by 2x-3 and combine like terms.

$$28x^{2}-65x+37-14x^{2}+63x-63-14x^{2}=5$$

Combine 28x^{2} and -14x^{2} to get 14x^{2}.

$$14x^{2}-65x+37+63x-63-14x^{2}=5$$

Combine -65x and 63x to get -2x.

$$14x^{2}-2x+37-63-14x^{2}=5$$

Subtract 63 from 37 to get -26.

$$14x^{2}-2x-26-14x^{2}=5$$

Combine 14x^{2} and -14x^{2} to get 0.

$$-2x-26=5$$

Add 26 to both sides.

$$-2x=5+26$$

Add 5 and 26 to get 31.

$$-2x=31$$

Divide both sides by -2.

$$x=\frac{31}{-2}$$

Fraction \frac{31}{-2} can be rewritten as -\frac{31}{2} by extracting the negative sign.

$$x=-\frac{31}{2}$$