1. **State the problem:** Find the equation of the circle passing through points $(1,-2)$ and $(4,-3)$ with its center on the line $3x + 4y = 7$. 2. **Formula and rules:** The general equation of a circle with center $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ The center $(h,k)$ lies on the line $3h + 4k = 7$. 3. **Use the points on the circle:** Since both points lie on the circle, their distances from the center are equal to the radius: $$ (1 - h)^2 + (-2 - k)^2 = r^2 $$ $$ (4 - h)^2 + (-3 - k)^2 = r^2 $$ 4. **Set the two expressions equal to eliminate $r^2$:** $$ (1 - h)^2 + (-2 - k)^2 = (4 - h)^2 + (-3 - k)^2 $$ 5. **Expand and simplify:** $$ (1 - h)^2 = (1 - 2h + h^2) $$ $$ (-2 - k)^2 = (4 + 4k + k^2) $$ $$ (4 - h)^2 = (16 - 8h + h^2) $$ $$ (-3 - k)^2 = (9 + 6k + k^2) $$ Substitute: $$ 1 - 2h + h^2 + 4 + 4k + k^2 = 16 - 8h + h^2 + 9 + 6k + k^2 $$ 6. **Cancel $h^2$ and $k^2$ on both sides:** $$ 5 - 2h + 4k = 25 - 8h + 6k $$ 7. **Bring all terms to one side:** $$ 5 - 2h + 4k - 25 + 8h - 6k = 0 $$ $$ (-20) + 6h - 2k = 0 $$ 8. **Simplify:** $$ 6h - 2k = 20 $$ Divide both sides by 2: $$ 3h - k = 10 $$ 9. **Use the line equation for the center:** $$ 3h + 4k = 7 $$ 10. **Solve the system:** From $3h - k = 10$, express $k$: $$ k = 3h - 10 $$ Substitute into $3h + 4k = 7$: $$ 3h + 4(3h - 10) = 7 $$ $$ 3h + 12h - 40 = 7 $$ $$ 15h = 47 $$ $$ h = \frac{47}{15} $$ 11. **Find $k$:** $$ k = 3 \times \frac{47}{15} - 10 = \frac{141}{15} - 10 = \frac{141}{15} - \frac{150}{15} = -\frac{9}{15} = -\frac{3}{5} $$ 12. **Calculate radius $r$ using point $(1,-2)$:** $$ r^2 = (1 - h)^2 + (-2 - k)^2 = \left(1 - \frac{47}{15}\right)^2 + \left(-2 + \frac{3}{5}\right)^2 $$ $$ = \left(-\frac{32}{15}\right)^2 + \left(-\frac{7}{5}\right)^2 = \frac{1024}{225} + \frac{49}{25} $$ Convert $\frac{49}{25}$ to denominator 225: $$ \frac{49}{25} = \frac{441}{225} $$ Sum: $$ r^2 = \frac{1024}{225} + \frac{441}{225} = \frac{1465}{225} $$ 13. **Write the equation of the circle:** $$ \left(x - \frac{47}{15}\right)^2 + \left(y + \frac{3}{5}\right)^2 = \frac{1465}{225} $$ **Final answer:** $$ \boxed{\left(x - \frac{47}{15}\right)^2 + \left(y + \frac{3}{5}\right)^2 = \frac{1465}{225}} $$