1. **State the problem:** We need to find which equation matches the line passing through points $(-8, -8)$ and $(8, 4)$.\n\n2. **Find the slope of the line:** The slope formula is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$\nSubstitute the points: $$m = \frac{4 - (-8)}{8 - (-8)} = \frac{12}{16} = \frac{3}{4}$$\nThe slope is positive $\frac{3}{4}$, consistent with the graph description.\n\n3. **Find the equation of the line in slope-intercept form $y = mx + b$:**\nUse point-slope form with point $(-8, -8)$:\n$$y - (-8) = \frac{3}{4}(x - (-8))$$\n$$y + 8 = \frac{3}{4}(x + 8)$$\n$$y + 8 = \frac{3}{4}x + 6$$\n$$y = \frac{3}{4}x + 6 - 8$$\n$$y = \frac{3}{4}x - 2$$\n\n4. **Convert to standard form $Ax + By = C$:**\nMultiply both sides by 4 to clear fractions:\n$$4y = 3x - 8$$\nRearranged:\n$$-3x + 4y = -8$$\nMultiply both sides by $-1$ to make $A$ positive:\n$$3x - 4y = 8$$\n\n5. **Check which given equation matches or is equivalent:**\n- $x - 8y = 24$ (No, coefficients don't match)\n- $-4x - 8y = 24$ (No)\n- $-8x + 4y = 24$ (No)\n- $4x - 8y = 24$ (Try dividing by 4: $x - 2y = 6$; does not match)\n\nNone exactly match $3x - 4y = 8$, but let's check if any are multiples:\nMultiply $3x - 4y = 8$ by 2:\n$$6x - 8y = 16$$\nNo given equation matches this.\n\n6. **Check the point $(0, -2)$ on each equation to see which fits the y-intercept:**\n- For $x - 8y = 24$: $0 - 8(-2) = 16 \neq 24$\n- For $-4x - 8y = 24$: $0 - 8(-2) = 16 \neq 24$\n- For $-8x + 4y = 24$: $0 + 4(-2) = -8 \neq 24$\n- For $4x - 8y = 24$: $0 - 8(-2) = 16 \neq 24$\n\n7. **Check the points given in the problem on each equation:**\nFor $(-8, -8)$:\n- $x - 8y = 24$: $-8 - 8(-8) = -8 + 64 = 56 \neq 24$\n- $-4x - 8y = 24$: $32 + 64 = 96 \neq 24$\n- $-8x + 4y = 24$: $64 - 32 = 32 \neq 24$\n- $4x - 8y = 24$: $-32 + 64 = 32 \neq 24$\n\n8. **Conclusion:** None of the given equations exactly match the line through $(-8, -8)$ and $(8, 4)$. The correct equation is $3x - 4y = 8$.\n\n**Final answer:** The line is represented by $$3x - 4y = 8$$ which is not among the given options.