Intercepts Equation F3103C

1. **Problem 1: Find the intercepts of the linear relationship given by the table:** Given table: $$\begin{array}{c|c} x & y \\ \hline 1 & 400 \\ 2 & 300 \\ 4 & 200 \\ 6 & 100 \\ 8 & 0 \end{array}$$ 2. **Find the x-intercept:** The x-intercept occurs when $$y=0$$. From the table, when $$y=0$$, $$x=8$$. So, the x-intercept is $$8$$. 3. **Find the y-intercept:** The y-intercept occurs when $$x=0$$. We don't have $$x=0$$ in the table, so we find the equation of the line first. 4. **Find the slope $$m$$:** Using points $$(1,400)$$ and $$(8,0)$$: $$m=\frac{0-400}{8-1}=\frac{-400}{7}=-\frac{400}{7}$$ 5. **Find the equation of the line:** Using point-slope form: $$y - y_1 = m(x - x_1)$$ Using point $$(1,400)$$: $$y - 400 = -\frac{400}{7}(x - 1)$$ Simplify: $$y = -\frac{400}{7}x + \frac{400}{7} + 400 = -\frac{400}{7}x + \frac{400}{7} + \frac{2800}{7} = -\frac{400}{7}x + \frac{3200}{7}$$ 6. **Find y-intercept by setting $$x=0$$:** $$y = -\frac{400}{7}(0) + \frac{3200}{7} = \frac{3200}{7} \approx 457.14$$ So, the y-intercept is approximately $$457.14$$. 7. **Interpretation of intercepts:** - The x-intercept $$8$$ means when $$x=8$$, $$y=0$$. - The y-intercept $$\approx 457.14$$ means when $$x=0$$, $$y$$ is about $$457.14$$. In context, if $$x$$ represents some quantity and $$y$$ another, these intercepts show where one variable is zero. --- 8. **Problem 2: Solve the equation $$3x + 2 = -1$$ by graphing and check the solution.** 9. **Rewrite the equation:** $$3x + 2 = -1$$ 10. **Subtract 2 from both sides:** $$3x + \cancel{2} - \cancel{2} = -1 - 2$$ $$3x = -3$$ 11. **Divide both sides by 3:** $$\frac{3x}{\cancel{3}} = \frac{-3}{\cancel{3}}$$ $$x = -1$$ 12. **Check the solution:** Substitute $$x = -1$$ into the original equation: $$3(-1) + 2 = -3 + 2 = -1$$ The left side equals the right side, so $$x = -1$$ is the solution.