1. **State the problem:** Solve the system of equations by graphing: $$x + y = 4$$ $$x - y = -2$$ 2. **Find intercepts for each line:** - For $$x + y = 4$$: - $$x$$-intercept: set $$y=0$$, then $$x=4$$, point is $$(4,0)$$ - $$y$$-intercept: set $$x=0$$, then $$y=4$$, point is $$(0,4)$$ - For $$x - y = -2$$: - $$x$$-intercept: set $$y=0$$, then $$x=-2$$, point is $$(-2,0)$$ - $$y$$-intercept: set $$x=0$$, then $$-y=-2 \Rightarrow y=2$$, point is $$(0,2)$$ 3. **Graph the lines using intercepts:** - Line 1 passes through $$(4,0)$$ and $$(0,4)$$. - Line 2 passes through $$(-2,0)$$ and $$(0,2)$$. 4. **Find the point of intersection:** Solve the system algebraically: Add the two equations: $$\begin{aligned} x + y &= 4 \\ x - y &= -2 \\ \hline 2x &= 2 \end{aligned}$$ Divide both sides by 2: $$2\cancel{x} = 2 \Rightarrow \cancel{2}x = 2 \Rightarrow x = 1$$ Substitute $$x=1$$ into $$x + y = 4$$: $$1 + y = 4 \Rightarrow y = 3$$ So the point of intersection is $$(1,3)$$. 5. **Check the solution:** Substitute $$(1,3)$$ into both equations: - $$1 + 3 = 4$$ (True) - $$1 - 3 = -2$$ (True) 6. **Interpretation:** - The lines intersect at one point. - This means the system is **consistent and independent**. - There is exactly one solution: $$(1,3)$$. --- **Final answer:** The solution to the system is $$(1,3)$$, and the system is consistent and independent with one unique solution.