Solve 10^ 4 -x 10^ 3-3x = x0 2X+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

$$\frac{10^{4}-x}{10^{3-3x}}=\frac{x0}{2X+2}}$$

Solve for X

$\left\{\begin{matrix}X=-\frac{x\times 1000^{x}+500x_{0}-10000\times 1000^{x}}{\left(x-10000\right)\times 10^{3x}}\text{, }&x_{0}\neq 0\text{ and }x\neq 10000\\X\neq -1\text{, }&x=10000\text{ and }x_{0}=0\end{matrix}\right.$

Steps for Solving Linear Equation

Variable $X$ cannot be equal to $-1$ since division by zero is not defined. Multiply both sides of the equation by $2\left(X+1\right)$.

$$\frac{1}{500}\left(X+1\right)\times 1000^{x}\left(10^{4}-x\right)=x_{0}$$

Use the distributive property to multiply $\frac{1}{500}$ by $X+1$.

$$\left(\frac{1}{500}X+\frac{1}{500}\right)\times 1000^{x}\left(10^{4}-x\right)=x_{0}$$

Use the distributive property to multiply $\frac{1}{500}X+\frac{1}{500}$ by $1000^{x}$.

$$\left(\frac{1}{500}X\times 1000^{x}+\frac{1}{500}\times 1000^{x}\right)\left(10^{4}-x\right)=x_{0}$$

Calculate $10$ to the power of $4$ and get $10000$.

$$\left(\frac{1}{500}X\times 1000^{x}+\frac{1}{500}\times 1000^{x}\right)\left(10000-x\right)=x_{0}$$

Use the distributive property to multiply $\frac{1}{500}X\times 1000^{x}+\frac{1}{500}\times 1000^{x}$ by $10000-x$.

$$20\times 1000^{x}X-\frac{1}{500}\times 1000^{x}xX+20\times 1000^{x}-\frac{1}{500}\times 1000^{x}x=x_{0}$$

Subtract $20\times 1000^{x}$ from both sides.

$$20\times 1000^{x}X-\frac{1}{500}\times 1000^{x}xX-\frac{1}{500}\times 1000^{x}x=x_{0}-20\times 1000^{x}$$

Add $\frac{1}{500}\times 1000^{x}x$ to both sides.

$$20\times 1000^{x}X-\frac{1}{500}\times 1000^{x}xX=x_{0}-20\times 1000^{x}+\frac{1}{500}\times 1000^{x}x$$

Combine all terms containing $X$.

$$\left(20\times 1000^{x}-\frac{1}{500}\times 1000^{x}x\right)X=x_{0}-20\times 1000^{x}+\frac{1}{500}\times 1000^{x}x$$

The equation is in standard form.

$$\left(-\frac{x\times 1000^{x}}{500}+20\times 1000^{x}\right)X=\frac{x\times 1000^{x}}{500}-20\times 1000^{x}+x_{0}$$

Divide both sides by $20\times 1000^{x}-\frac{1}{500}\times 1000^{x}x$.

$$\frac{\left(-\frac{x\times 1000^{x}}{500}+20\times 1000^{x}\right)X}{-\frac{x\times 1000^{x}}{500}+20\times 1000^{x}}=\frac{\frac{x\times 1000^{x}}{500}-20\times 1000^{x}+x_{0}}{-\frac{x\times 1000^{x}}{500}+20\times 1000^{x}}$$

Dividing by $20\times 1000^{x}-\frac{1}{500}\times 1000^{x}x$ undoes the multiplication by $20\times 1000^{x}-\frac{1}{500}\times 1000^{x}x$.

$$X=\frac{\frac{x\times 1000^{x}}{500}-20\times 1000^{x}+x_{0}}{-\frac{x\times 1000^{x}}{500}+20\times 1000^{x}}$$

Divide $x_{0}-20\times 1000^{x}+\frac{1000^{x}x}{500}$ by $20\times 1000^{x}-\frac{1}{500}\times 1000^{x}x$.

$$X=\frac{x\times 1000^{x}+500x_{0}-10000\times 1000^{x}}{\left(10000-x\right)\times 10^{3x}}$$

Variable $X$ cannot be equal to $-1$.

$$X=\frac{x\times 1000^{x}+500x_{0}-10000\times 1000^{x}}{\left(10000-x\right)\times 10^{3x}}\text{, }X\neq -1$$