1. **State the problem:** We are given three pairs of values $(x, y)$ involving a constant $s$ and told there is a linear relationship between $x$ and $y$. We need to find which equation among the options represents this relationship. 2. **List the points:** $$ (-2s, 24), \quad (-s, 21), \quad (s, 15) $$ 3. **Check each equation option by substituting the points:** - Option A: $sx + 3y = 18s$ - For $x = -2s$, $y = 24$: $$s(-2s) + 3(24) = -2s^2 + 72$$ This should equal $18s$, so: $$-2s^2 + 72 = 18s$$ This is not generally true for all $s$. - Option B: $3x + sy = 18s$ - For $x = -2s$, $y = 24$: $$3(-2s) + s(24) = -6s + 24s = 18s$$ This matches perfectly. - Check for $x = -s$, $y = 21$: $$3(-s) + s(21) = -3s + 21s = 18s$$ Matches again. - Check for $x = s$, $y = 15$: $$3(s) + s(15) = 3s + 15s = 18s$$ Matches again. - Option C: $3x + sy = 18$ - For $x = -2s$, $y = 24$: $$3(-2s) + s(24) = -6s + 24s = 18s$$ This equals $18s$, not $18$, so this option is false. - Option D: $sx + 3y = 18$ - For $x = -2s$, $y = 24$: $$s(-2s) + 3(24) = -2s^2 + 72$$ This equals $18$ only if $-2s^2 + 72 = 18$, which is not true for all $s$. 4. **Conclusion:** Only option B satisfies the linear relationship for all given points. **Final answer:** $$3x + sy = 18s$$