Find parametric equations for the line of intersection of the planes 2x - y + 3z = 5 and 3x - 2y - z = 7

Solution:

Given equations are

2x - y + 3z = 5

3x - 2y - z = 7

Let us consider z = t

2x - y + 3t = 5

3x - 2y - t = 7

We can write it as

2x - y = 5 - 3t ….. (1)

3x - 2y = 7 + t …. (2)

Multiply equation (1) by 2 - equation (2)

4x - 2y - (3x - 2y) = 10 - 6t - (7 + t)

4x - 2y - 3x + 2y = 10 - 6t - 7 - t

x = 3 - 7t

Let us substitute the value of x in equation (1)

2 (3 - 7t) - y = 5 - 3t

6 - 14t - y = 5 - 3t

6 - 5 - y = -3t + 14t

11t = 1 - y

y = 1 - 11t

Therefore, the parametric equations for the line are x = 3 - 7t, y = 1 - 11t and z = t.

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Find parametric equations for the line of intersection of the planes 2x - y + 3z = 5 and 3x - 2y - z = 7

Summary:

The parametric equations for the line of intersection of the planes 2x - y + 3z = 5 and 3x - 2y - z = 7 is x = 3 - 7t, y = 1 - 11t and z = t.