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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.