Solve x-5/100+x-4/101+x-3/102=x-100/5+frac{x-101}{4}+x

$$\frac { x - 5 } { 100 } + \frac { x - 4 } { 101 } + \frac { x - 3 } { 102 } = \frac { x - 100 } { 5 } + \frac { x - 101 } { 4 } + x$$

Solve for x

$$x = \frac{11623485}{365797} = 31\frac{283778}{365797} \approx 31.775780009$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 515100, the least common multiple of 100,101,102,5,4.

$$5151\left(x-5\right)+5100\left(x-4\right)+5050\left(x-3\right)=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Use the distributive property to multiply 5151 by x-5.

$$5151x-25755+5100\left(x-4\right)+5050\left(x-3\right)=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Use the distributive property to multiply 5100 by x-4.

$$5151x-25755+5100x-20400+5050\left(x-3\right)=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Combine 5151x and 5100x to get 10251x.

$$10251x-25755-20400+5050\left(x-3\right)=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Subtract 20400 from -25755 to get -46155.

$$10251x-46155+5050\left(x-3\right)=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Use the distributive property to multiply 5050 by x-3.

$$10251x-46155+5050x-15150=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Combine 10251x and 5050x to get 15301x.

$$15301x-46155-15150=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Subtract 15150 from -46155 to get -61305.

$$15301x-61305=103020\left(x-100\right)+128775\left(x-101\right)+515100x$$

Use the distributive property to multiply 103020 by x-100.

$$15301x-61305=103020x-10302000+128775\left(x-101\right)+515100x$$

Use the distributive property to multiply 128775 by x-101.

$$15301x-61305=103020x-10302000+128775x-13006275+515100x$$

Combine 103020x and 128775x to get 231795x.

$$15301x-61305=231795x-10302000-13006275+515100x$$

Subtract 13006275 from -10302000 to get -23308275.

$$15301x-61305=231795x-23308275+515100x$$

Combine 231795x and 515100x to get 746895x.

$$15301x-61305=746895x-23308275$$

Subtract 746895x from both sides.

$$15301x-61305-746895x=-23308275$$

Combine 15301x and -746895x to get -731594x.

$$-731594x-61305=-23308275$$

Add 61305 to both sides.

$$-731594x=-23308275+61305$$

Add -23308275 and 61305 to get -23246970.

$$-731594x=-23246970$$

Divide both sides by -731594.

$$x=\frac{-23246970}{-731594}$$

Reduce the fraction \frac{-23246970}{-731594} to lowest terms by extracting and canceling out -2.

$$x=\frac{11623485}{365797}$$