The equation 4x – 3y = 12 describes a line in 3D space

The equation 4x – 3y = 12 describes a line in 3D space

– True

– False

The correct answer and explanation is:

The statement is False.

Here’s why:

The equation 4x−3y=124x – 3y = 12 is a linear equation involving two variables, xx and yy. This equation describes a straight line, but only in a two-dimensional (2D) plane, not in 3D space. In a 2D space, a linear equation like this one represents a straight line, where each point on the line is a solution to the equation.

In contrast, a line in 3D space requires a different type of equation. For a line to be represented in 3D space, it typically needs a parametric equation or two equations that describe both xx, yy, and zz coordinates. For example, a parametric equation for a line in 3D could look like this: x=x0+t⋅vxx = x_0 + t \cdot v_x y=y0+t⋅vyy = y_0 + t \cdot v_y z=z0+t⋅vzz = z_0 + t \cdot v_z

Here, (x0,y0,z0)(x_0, y_0, z_0) is a point on the line, and (vx,vy,vz)(v_x, v_y, v_z) is a direction vector. The parameter tt allows you to get all points on the line in 3D space.

The equation 4x−3y=124x – 3y = 12 describes a line in 2D because there is no zz-coordinate involved. If you wanted this to describe a line in 3D, you would need to add a zz-coordinate (or even better, specify another equation relating xx, yy, and zz).

So, to summarize, the equation only represents a line in 2D, not in 3D space. Therefore, the correct answer is False.