1. **Problem Statement:** Find the gradient (slope) and y-intercept of the line given by the equation $$2x + 3y = 3$$ and sketch the graph. 2. **Formula and Rules:** - The gradient (slope) $$m$$ of a line in the form $$y = mx + c$$ is the coefficient of $$x$$. - The y-intercept $$c$$ is the value of $$y$$ when $$x = 0$$. - To find $$m$$ and $$c$$, rewrite the equation in the form $$y = mx + c$$. 3. **Rewrite the equation:** $$2x + 3y = 3$$ Subtract $$2x$$ from both sides: $$3y = 3 - 2x$$ Divide both sides by 3: $$y = \frac{3 - 2x}{3}$$ Show cancellation: $$y = \frac{\cancel{3} - 2x}{\cancel{3}} = 1 - \frac{2}{3}x$$ Rearranged as: $$y = -\frac{2}{3}x + 1$$ 4. **Identify gradient and y-intercept:** - Gradient $$m = -\frac{2}{3}$$ - Y-intercept $$c = 1$$ 5. **Interpretation:** - The line slopes downward because the gradient is negative. - The line crosses the y-axis at $$y = 1$$. 6. **Find x-intercept:** Set $$y = 0$$: $$0 = -\frac{2}{3}x + 1$$ Solve for $$x$$: $$\frac{2}{3}x = 1$$ $$x = \frac{1}{\frac{2}{3}} = \frac{1 \times 3}{2} = \frac{3}{2} = 1.5$$ 7. **Summary:** - Gradient: $$-\frac{2}{3}$$ - Y-intercept: $1$ - X-intercept: $1.5$ 8. **Sketching:** - Plot the y-intercept at $(0,1)$. - Plot the x-intercept at $(1.5,0)$. - Draw a straight line through these points. This completes the solution.