1. **Stating the problem:** We are given the equation of a line: $2x + 3y + 4 = 10$. 2. **Rewrite the equation in slope-intercept form:** We want to express $y$ in terms of $x$ to find the slope and intercepts. Start with: $$2x + 3y + 4 = 10$$ Subtract 4 from both sides: $$2x + 3y = 6$$ 3. **Isolate $y$:** $$3y = 6 - 2x$$ Divide both sides by 3: $$y = \frac{6 - 2x}{3}$$ Show cancellation: $$y = \frac{\cancel{6} - 2x}{\cancel{3}}$$ Simplify: $$y = 2 - \frac{2}{3}x$$ 4. **Identify slope and intercept:** The equation is now in the form $y = mx + b$ where: - Slope $m = -\frac{2}{3}$ - $y$-intercept $b = 2$ 5. **Interpret the slope:** The slope $-\frac{2}{3}$ means for every increase of 3 units in $x$, $y$ decreases by 2 units. 6. **Check the gradient calculation:** The handwritten note says $4 \div 2 = 2$, which is the gradient between points $(0, -2)$ and $(4, 2)$. Calculate slope between these points: $$m = \frac{2 - (-2)}{4 - 0} = \frac{4}{4} = 1$$ This slope is $1$, which does not match the slope from the line equation $-\frac{2}{3}$. 7. **Conclusion:** The line $2x + 3y + 4 = 10$ has slope $-\frac{2}{3}$ and $y$-intercept $2$. The handwritten gradient $4 \div 2 = 2$ does not correspond to this line's slope. **Final answer:** $$y = 2 - \frac{2}{3}x$$ Slope $m = -\frac{2}{3}$, $y$-intercept $2$.