1. **State the problem:** We have a linear equation with points $(0,a)$, $(2,b)$, and $(6,8)$ on its graph. 2. **Recall the formula for the rate of change (slope) of a linear function:** $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. Since the points lie on the same line, the slope between any two points must be equal. 4. Calculate the slope between $(0,a)$ and $(6,8)$: $$m = \frac{8 - a}{6 - 0} = \frac{8 - a}{6}$$ 5. Calculate the slope between $(2,b)$ and $(6,8)$: $$m = \frac{8 - b}{6 - 2} = \frac{8 - b}{4}$$ 6. Set the slopes equal because the line is linear: $$\frac{8 - a}{6} = \frac{8 - b}{4}$$ 7. Cross multiply: $$4(8 - a) = 6(8 - b)$$ 8. Expand both sides: $$32 - 4a = 48 - 6b$$ 9. Rearrange to isolate variables: $$-4a + 6b = 48 - 32$$ $$-4a + 6b = 16$$ 10. Simplify by dividing the entire equation by 2: $$\cancel{\frac{-4a}{2}} + \cancel{\frac{6b}{2}} = \frac{16}{2}$$ $$-2a + 3b = 8$$ 11. Express $b$ in terms of $a$: $$3b = 8 + 2a$$ $$b = \frac{8 + 2a}{3}$$ **Final answer:** $$a = a \text{ (any real number)}$$ $$b = \frac{8 + 2a}{3}$$ This means $a$ can be any real number, and $b$ depends on $a$ as shown.