Let's break this down step by step:

Part 4: Write Equations from Different Representations

1. Rewriting $y - 3 = 2(x - 1)$ in slope-intercept form: Start with the given equation: $y - 3 = 2(x - 1)$ Expand $2(x - 1)$: $y - 3 = 2x - 2$ Add 3 to both sides: $y = 2x + 1$ Slope-intercept form: $y = 2x + 1$.

2. Equation of the line given $m = -4$ and point $(-2, -6)$: The point-slope form is: $y - y_1 = m(x - x_1)$ Substituting $m = -4$ and $(x_1, y_1) = (-2, -6)$: $y - (-6) = -4(x - (-2))$ Simplify: $y + 6 = -4(x + 2)$ Expand: $y + 6 = -4x - 8$ Subtract 6 from both sides: $y = -4x - 14$ Point-slope form: $y + 6 = -4(x + 2)$, Standard form: $y = -4x - 14$.

3. Equation of the line through $(0,0)$ with $m = \frac{5}{2}$: Using the slope-intercept form $y = mx + b$, substitute $m = \frac{5}{2}$ and the point $(0,0)$: $y = \frac{5}{2}x + 0$ $y = \frac{5}{2}x$ Equation: $y = \frac{5}{2}x$.

Challenge Question

1. Line 1: Passes through $(7, 5)$ with slope $-3$: Using point-slope form: $y - 5 = -3(x - 7)$ Expand: $y - 5 = -3x + 21$ Add 5: $y = -3x + 26$

2. Line 2: Passes through $(7, 5)$ with slope $\frac{3}{4}$: Using point-slope form: $y - 5 = \frac{3}{4}(x - 7)$ Expand: $y - 5 = \frac{3}{4}x - \frac{21}{4}$ Add 5 (or $\frac{20}{4}$): $y = \frac{3}{4}x + \frac{-21 + 20}{4}$ $y = \frac{3}{4}x - \frac{1}{4}$

3. Checking perpendicularity: Slopes of perpendicular lines multiply to $-1$: $(-3) \cdot \frac{3}{4} = -\frac{9}{4}$ Since this is not $-1$, the lines are not perpendicular.

Would you like further clarifications or more details? Here are 5 related questions:

1. How do you determine the slope of a line from two given points?
2. What is the standard form of a linear equation, and how is it derived from slope-intercept form?
3. How do you verify if two lines are parallel?
4. Can a slope be undefined? If so, what does it represent?
5. What are the key differences between point-slope and slope-intercept forms?

Tip: Always start with slope-intercept form for simplicity, and convert to other forms as needed.