Write the canonical and parametric equations of the straight line given by the general equations: 4x + 2y

univerkov | March 10, 2021 | education

Write the canonical and parametric equations of the straight line given by the general equations: 4x + 2y + 3z + 2 = 0; 4x + 3y + 4z + 1 = 0

The intersection of two non-parallel planes define the equation of a straight line.

4 x + 2 y + 3 z + 2 = 0;

4 x + 3 y + 4 z + 1 = 0.

Let’s rewrite the system of equations.

4 x + 2 y = – 3 z – 2;

4 x + 3 y = – 4 z – 1.

Subtract the first equation from the second equation:

y = – z + 1;

Substitute this y value in the first equation:

4 x + 2 (- z + 1) = – 3 z – 2;

4 x + 2 (- z + 1) = – z – 4;

x = – z / 4 – 1.

Let us write the equation of a straight line defined parametrically in t.

x = – t / 4 – 1.

y = – t + 1.

z = t.

Find t for each equation:

t = – 4 x – 4;

t = – y + 1;

t = z;

The canonical equation of the straight line:

– 4 x – 4 = – y + 1 = z The intersection of two non-parallel planes defines the equation of a straight line.