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Section 12.2 : Equations of Lines

7. Does the line given by $$x = 9 + 21t$$, $$y = - 7$$, $$z = 12 - 11t$$ intersect the $$xz$$-plane? If so, give the point.

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If the line intersects the $$xz$$-plane there will be a point on the line that is also in the $$xz$$‑plane. Recall as well that any point in the $$xz$$-plane will have a $$y$$ coordinate of $$y = 0$$.

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So, to determine if the line intersects the $$xz$$-plane all we need to do is set the equation for the $$y$$ coordinate equal to zero and solve it for $$t$$, if that’s possible.

However, in this case we can see that is clearly not possible since the $$y$$ equation is simply $$y = - 7$$ and this can clearly never be zero.

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Therefore, the line does not intersect the $$xz$$-plane.