Solve for X in -x + 8/3x + 5 = 1-x: Linear Equation with Fractions

To solve this linear equation, we will follow these steps:

• Simplify the left-hand side by expanding and combining like terms.
• Isolate $x$ on one side of the equation.
• Solve for $x$.

Let's perform these steps:

We start with the equation: $-x + 8 \cdot \frac{1}{3}x + 5 = 1 - x$.

First, simplify the term $8 \cdot \frac{1}{3}x$ to $\frac{8}{3}x$.

The equation becomes:

$-x + \frac{8}{3}x + 5 = 1 - x$.

Combine the like terms $-x$ and $\frac{8}{3}x$:

$\left(-1 + \frac{8}{3}\right)x = \frac{8}{3}x - \frac{3}{3}x = \frac{5}{3}x$.

The equation simplifies to:

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

Add $x$ to both sides to eliminate $x$ on the right-hand side:

$\frac{5}{3}x + x + 5 = 1$.

Convert $x$ to a fraction: $x = \frac{3}{3}x$, so:

$\left(\frac{5}{3} + \frac{3}{3}\right)x + 5 = 1$.

This simplifies to:

$\frac{8}{3}x + 5 = 1$.

Subtract $5$ from both sides to isolate terms involving $x$:

$\frac{8}{3}x = 1 - 5$, which simplifies to:

$\frac{8}{3}x = -4$.

Isolate $x$ by multiplying both sides by the reciprocal of $\frac{8}{3}$:

$x = -4 \times \frac{3}{8}$.

Calculate the value:

$x = -\frac{12}{8}$.

Simplify the fraction:

$x = -\frac{3}{2}$.

Therefore, the solution to the equation is $x = -\frac{3}{2}$.