1. The problem is to verify if the equation manipulation from $2x + 3y = 12$ to $y = \frac{2}{3}x - 4$ is correct. 2. Start with the original equation: $$2x + 3y = 12$$ 3. Subtract $2x$ from both sides: $$2x + 3y - 2x = 12 - 2x$$ $$\cancel{2x} + 3y - \cancel{2x} = 12 - 2x$$ $$3y = 12 - 2x$$ 4. Divide both sides by 3 to solve for $y$: $$y = \frac{12 - 2x}{3}$$ $$y = \frac{12}{3} - \frac{2x}{3}$$ $$y = 4 - \frac{2}{3}x$$ 5. The final expression is: $$y = 4 - \frac{2}{3}x$$ 6. The expression you wrote, $y = \frac{2}{3}x - 4$, is not correct because the signs are reversed. 7. So the correct slope-intercept form is: $$y = -\frac{2}{3}x + 4$$ This means the slope is $-\frac{2}{3}$ and the y-intercept is 4, not $\frac{2}{3}$ and $-4$ as in your expression.