曲线方程为 $y = \frac{5x}{6 - 2x}$。求曲线斜率为 7.5 的点的 $x$ 坐标。

1. Find the derivative of the curve using the quotient rule

Let $y = \frac{5x}{6 - 2x}$.
Using the quotient rule:
$\frac{dy}{dx} = \frac{(6 - 2x)\frac{d}{dx}(5x) - (5x)\frac{d}{dx}(6 - 2x)}{(6 - 2x)^2}$
$\frac{dy}{dx} = \frac{(6 - 2x)(5) - (5x)(-2)}{(6 - 2x)^2}$

2. Simplify the derivative

$\frac{dy}{dx} = \frac{30 - 10x + 10x}{(6 - 2x)^2} = \frac{30}{(6 - 2x)^2}$

3. Set the gradient to 7.5 and solve for $x$

Given the gradient is 7.5:
$\frac{30}{(6 - 2x)^2} = 7.5$
$(6 - 2x)^2 = \frac{30}{7.5}$
$(6 - 2x)^2 = 4$
$6 - 2x = 2 \quad \text{or} \quad 6 - 2x = -2$
$2x = 4 \quad \text{or} \quad 2x = 8$
$x = 2 \quad \text{or} \quad x = 4$