Solve 2/(x - 3) + 1/(x - 1) = 5/(x - 1) - 2/(x - 2) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{2}{x-3}+\frac{1}{x-1}=\frac{5}{x-1}-\frac{2}{x-2}$$

Solve for x

$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$

Steps for Solving Linear Equation

Variable $x$ cannot be equal to any of the values $1,2,3$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-3\right)\left(x-2\right)\left(x-1\right)$, the least common multiple of $x-3,x-1,x-2$.

$$\left(x-2\right)\left(x-1\right)\times 2+\left(x-3\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)\times 2$$

Use the distributive property to multiply $x-2$ by $x-1$ and combine like terms.

$$\left(x^{2}-3x+2\right)\times 2+\left(x-3\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)\times 2$$

Use the distributive property to multiply $x^{2}-3x+2$ by $2$.

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

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

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

Combine $2x^{2}$ and $x^{2}$ to get $3x^{2}$.

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

Combine $-6x$ and $-5x$ to get $-11x$.

$$3x^{2}-11x+4+6=\left(x-3\right)\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)\times 2$$

Add $4$ and $6$ to get $10$.

$$3x^{2}-11x+10=\left(x-3\right)\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)\times 2$$

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

$$3x^{2}-11x+10=\left(x^{2}-5x+6\right)\times 5-\left(x-3\right)\left(x-1\right)\times 2$$

Use the distributive property to multiply $x^{2}-5x+6$ by $5$.

$$3x^{2}-11x+10=5x^{2}-25x+30-\left(x-3\right)\left(x-1\right)\times 2$$

Use the distributive property to multiply $x-3$ by $x-1$ and combine like terms.

$$3x^{2}-11x+10=5x^{2}-25x+30-\left(x^{2}-4x+3\right)\times 2$$

Use the distributive property to multiply $x^{2}-4x+3$ by $2$.

$$3x^{2}-11x+10=5x^{2}-25x+30-\left(2x^{2}-8x+6\right)$$

To find the opposite of $2x^{2}-8x+6$, find the opposite of each term.

$$3x^{2}-11x+10=5x^{2}-25x+30-2x^{2}+8x-6$$

Combine $5x^{2}$ and $-2x^{2}$ to get $3x^{2}$.

$$3x^{2}-11x+10=3x^{2}-25x+30+8x-6$$

Combine $-25x$ and $8x$ to get $-17x$.

$$3x^{2}-11x+10=3x^{2}-17x+30-6$$

Subtract $6$ from $30$ to get $24$.

$$3x^{2}-11x+10=3x^{2}-17x+24$$

Subtract $3x^{2}$ from both sides.

$$3x^{2}-11x+10-3x^{2}=-17x+24$$

Combine $3x^{2}$ and $-3x^{2}$ to get $0$.

$$-11x+10=-17x+24$$

Add $17x$ to both sides.

$$-11x+10+17x=24$$

Combine $-11x$ and $17x$ to get $6x$.

$$6x+10=24$$

Subtract $10$ from both sides.

$$6x=24-10$$

Subtract $10$ from $24$ to get $14$.

$$6x=14$$

Divide both sides by $6$.

$$x=\frac{14}{6}$$

Reduce the fraction $\frac{14}{6}$ to lowest terms by extracting and canceling out $2$.

$$x=\frac{7}{3}$$