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Answer :
Final answer:
The symmetric equations that represent the line given by the parametric equations are found by setting the ratios for the parameter t equal to each other. The correct set of symmetric equations is (x + 1)/15 = -y - 7/1 = (z - 15)/(-15), which corresponds to option (b).
Explanation:
The question involves finding a set of symmetric equations that represent a given line with parametric equations (x = -1 + 15t), (y = -7), (z = 15 - 15t). Symmetric equations give a relationship between x, y, and z that defines the line in a non-parametric form and are typically written with the free parameter eliminated, showing the proportionality between the subtracted coordinates and their respective direction numbers.
To find the symmetric equations, we look for ratios that equate to the parameter t. From the given parametric equations, for x, we have x = -1 + 15t, which rearranges to t = (x + 1)/15. For z, we have z = 15 - 15t, rearranging to t = (15 - z)/15. Since y is a constant (-7), its equation simplifies to y = -7.
The corresponding value of t from x and z should be equal for all points on the line, so we set (x + 1)/15 = (15 - z)/15, which simplifies to (x + 1)/15 = (z - 15)/(-15). Because the y-component is a constant, its symmetric equation doesn't share the same denominator and is independent of t. Thus, the symmetric equations are (x + 1)/15 = (z - 15)/(-15), with y = -7 being a separate equation.
Looking at the options provided in the question, the correct symmetric equation that matches our derived forms is (x + 1)/15 = -y - 7/1 = (z - 15)/(-15), since -y - 7 simplifies to y = -7. Therefore, the correct option is (b): (x + 1)/15 = -y - 7 = (z - 15)/(-15).
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