Solve for X: Rectangle Perimeter 18 with Side Lengths 5x and 10x-8

To solve this problem, we'll follow these steps:

• Step 1: Write the expression for the perimeter of rectangle GCHF using the given variables.
• Step 2: Set up the equation by equating the expression to 18.
• Step 3: Solve the equation for $x$.

Let's work through each step:

Step 1: The perimeter of rectangle GCHF is given by the formula:

$\text{Perimeter} = 2 \times (\text{length}) + 2 \times (\text{width})$

According to the diagram, the dimensions of the rectangle GCHF are:

- GH = $3 + x$ (length),

- GF = $5x$ (width).

Thus, the perimeter formula becomes:

$2 \times (3 + x) + 2 \times (5x)$

Step 2: Set the equation equal to the given perimeter, 18:

$2(3 + x) + 2(5x) = 18$

Step 3: Simplify and solve for $x$:

Distribute the 2:

$6 + 2x + 10x = 18$

Combine like terms:

$6 + 12x = 18$

Subtract 6 from both sides:

$12x = 12$

Divide both sides by 12 to isolate $x$:

$x = 1$

Therefore, the solution to the problem is $x = 1$.