Question

Label any intercepts and sketch a graph of the plane. $$ 2 x-y+3 z=4 $$

Short answer

The x-intercept on the plane is at (2, 0, 0), the y-intercept at (0, -4, 0) and the z-intercept at (0, 0, \frac{4}{3}).

Step-by-step solution

Find the x-intercept

Setting y and z to zero, and solving for x gives the x-intercept. So, let's put y = 0 and z = 0 in the equation \(2x - y + 3z = 4\). This simplifies to \(2x = 4\) and solving for x gives \(x = 2\). So, the x-intercept is (2, 0, 0).

Find the y-intercept

Setting x and z to zero, and solving for y gives the y-intercept. So, let's plug x = 0 and z = 0 in the equation \(2x - y + 3z = 4\). This simplifies to \(-y = 4\) and solving for y gives \(y = -4\). So, the y-intercept is (0, -4, 0).

Find the z-intercept

Setting x and y to zero, and solving for z gives the z-intercept. Plugging x = 0 and y = 0 into the equation \(2x - y + 3z = 4\) simplifies to \(3z = 4\). Solving for z gives \(z = \frac{4}{3}\). So, the z-intercept is (0, 0, \frac{4}{3}).

Sketch the graph

In a three dimensional plane, plot the x-intercept at (2, 0, 0), the y-intercept at (0, -4, 0) and the z-intercept at (0, 0, \frac{4}{3}). Connect the three points to form a plane.