WRITING AN EQUATION Write an equation in point-slope form of the line that passes through the given point and has the given slope. \( \begin{array}{lll}\text { 8. }(3,0) ; m=-\frac{2}{3} & \text { 9. }(4,8) ; m=\frac{3}{4} & \text { 10. }(1,-3) ; m=4\end{array} \) \( \begin{array}{lll}\text { 11. }(7,-5) ; m=-\frac{1}{7} & \text { 12. }(3,3) ; m=\frac{5}{3} & \text { 13. }(-1,-4) ; m=-2\end{array} \)

To transform the point-slope formula into equations for the given points and slopes, you can use the formula \( y - y_1 = m(x - x_1) \) where \( (x_1, y_1) \) is the point and \( m \) is the slope. For point \( (3,0) \) with slope \( -\frac{2}{3} \), the equation is: \( y - 0 = -\frac{2}{3}(x - 3) \) or \( y = -\frac{2}{3}(x - 3) \). For point \( (4,8) \) with slope \( \frac{3}{4} \), the equation is: \( y - 8 = \frac{3}{4}(x - 4) \) or \( y = \frac{3}{4}(x - 4) + 8 \). For point \( (1,-3) \) with slope \( 4 \), the equation is: \( y + 3 = 4(x - 1) \) or \( y = 4(x - 1) - 3 \). And for point \( (7,-5) \) with slope \( -\frac{1}{7} \), the equation is: \( y + 5 = -\frac{1}{7}(x - 7) \) or \( y = -\frac{1}{7}(x - 7) - 5 \). You can apply the same formula to the other points provided for their respective slopes. Understanding the historical evolution of algebra enhances appreciation of its concepts. Algebra as we know it has roots in ancient Babylon, where mathematicians used geometric methods. Fast forward to the 9th century, the Persian mathematician Al-Khwarizmi laid the groundwork for algebraic symbolism and techniques that paved the way for modern algebra! When tackling point-slope equations, watch out for common mistakes! A frequent pitfall is mixing up the signs when substituting the slope into the equation. Remember to keep your slope as \( m \) and substitute that directly while wrapping your mind around the negative signs from \( (y_1) \) and \( (x_1) \). Double-checking your slope and point coordinates can save you from working with the wrong version of the equation!