1. **Problem:** Solve the system graphically for part (a): $$y = -3x + 9$$ $$3y - 3x = 3$$ 2. **Rewrite the second equation in slope-intercept form:** $$3y - 3x = 3 \implies 3y = 3x + 3 \implies y = \frac{3x + 3}{3} = x + 1$$ 3. **Plot both lines:** - Line 1: $y = -3x + 9$ - Line 2: $y = x + 1$ 4. **Find the point of intersection (POI) algebraically:** Set $-3x + 9 = x + 1$ $$-3x + 9 = x + 1$$ $$-3x - x = 1 - 9$$ $$-4x = -8$$ $$x = \frac{\cancel{-8}}{\cancel{-4}} = 2$$ Substitute $x=2$ into $y = x + 1$: $$y = 2 + 1 = 3$$ POI is $(2, 3)$. --- 1. **Problem:** Solve the system graphically for part (b): $$y = -2x + 4$$ $$3y - 2x + 12 = 0$$ 2. **Rewrite the second equation in slope-intercept form:** $$3y - 2x + 12 = 0 \implies 3y = 2x - 12 \implies y = \frac{2x - 12}{3} = \frac{2}{3}x - 4$$ 3. **Plot both lines:** - Line 1: $y = -2x + 4$ - Line 2: $y = \frac{2}{3}x - 4$ 4. **Find the POI algebraically:** Set $-2x + 4 = \frac{2}{3}x - 4$ Multiply both sides by 3 to clear denominator: $$3(-2x + 4) = 3\left(\frac{2}{3}x - 4\right)$$ $$-6x + 12 = 2x - 12$$ Bring terms to one side: $$-6x - 2x = -12 - 12$$ $$-8x = -24$$ $$x = \frac{\cancel{-24}}{\cancel{-8}} = 3$$ Substitute $x=3$ into $y = -2x + 4$: $$y = -2(3) + 4 = -6 + 4 = -2$$ POI is $(3, -2)$. --- 1. **Problem:** Solve the system graphically for part (c): $$4y + 4 = -3x$$ $$3y + 9x + 3 = 0$$ 2. **Rewrite both equations in slope-intercept form:** First equation: $$4y + 4 = -3x \implies 4y = -3x - 4 \implies y = \frac{-3x - 4}{4} = -\frac{3}{4}x - 1$$ Second equation: $$3y + 9x + 3 = 0 \implies 3y = -9x - 3 \implies y = \frac{-9x - 3}{3} = -3x - 1$$ 3. **Plot both lines:** - Line 1: $y = -\frac{3}{4}x - 1$ - Line 2: $y = -3x - 1$ 4. **Find the POI algebraically:** Set $-\frac{3}{4}x - 1 = -3x - 1$ Add 1 to both sides: $$-\frac{3}{4}x = -3x$$ Multiply both sides by 4 to clear denominator: $$-3x = -12x$$ Add $12x$ to both sides: $$-3x + 12x = 0 \implies 9x = 0 \implies x = 0$$ Substitute $x=0$ into $y = -3x - 1$: $$y = -3(0) - 1 = -1$$ POI is $(0, -1)$. --- **Final answers:** - (a) POI: $(2, 3)$ - (b) POI: $(3, -2)$ - (c) POI: $(0, -1)$