Question

Give any two points in the given plane. \(3(x+2)+5(y-9)-4 z=0\)

Short answer

Points (0, 9, 1.5) and (1, 10, 3.5) lie in the plane.

Step-by-step solution

Simplify the Equation

First, let's simplify the equation by distributing the numbers: \[ 3(x+2) + 5(y-9) - 4z = 0 \] simplifies to \[ 3x + 6 + 5y - 45 - 4z = 0 \]. Simplifying further, this becomes: \[ 3x + 5y - 4z - 39 = 0 \].

Choose a Value for x

To find one point, let’s choose a simple value for \(x\), such as \(x = 0\). This will help us easily solve for the other coordinates, \(y\) and \(z\).

Solve for y and z with x=0

Substitute \(x = 0\) into the equation: \[ 3(0) + 5y - 4z - 39 = 0 \] simplifies to \[ 5y - 4z - 39 = 0 \]. Now we can choose a simple value for \(y\) (e.g., \(y = 9\)) and solve for \(z\): \[ 5(9) - 4z - 39 = 0 \] simplifies to \[ 45 - 4z - 39 = 0 \], which gives us \(4z = 6\), so \(z = 1.5\). One point is \((0, 9, 1.5)\).

Choose Another Value for x

For the second point, choose another value for \(x\), such as \(x = 1\).

Solve for y and z with x=1

With \(x = 1\), substitute back into the equation: \[ 3(1) + 5y - 4z - 39 = 0 \] simplifies to \[ 3 + 5y - 4z - 39 = 0 \], then \[ 5y - 4z - 36 = 0 \]. Choose \(y = 10\) to solve for \(z\): \[ 5(10) - 4z - 36 = 0 \] simplifies to \[ 50 - 4z - 36 = 0 \], leading to \(4z = 14\), and \(z = 3.5\). The second point is \((1, 10, 3.5)\).

Key concepts

Planes in Three-Dimensions

In three-dimensional space, a plane is like an infinite flat sheet. To define this plane analytically, you need an equation that represents all the points on it. Planes are usually described using a linear equation of the form \( Ax + By + Cz = D \), where \( A \), \( B \), \( C \), and \( D \) are constants. These constants define the orientation and position of the plane in space.
To understand the equation of a plane, imagine it as a rule that every point \((x, y, z)\) on the plane must satisfy. The normal vector to the plane, given by \( (A, B, C) \), is crucial because it is perpendicular to the surface of the plane. This perpendicular relationship helps in various calculations like finding angles, distances, and intersections with other geometric structures.
When working with planes, it's essential to visualize the arrangement in space and how the coefficients in the equation determine its tilt and direction. This understanding is foundational in fields like physics, computer graphics, and engineering where three-dimensional modeling is key.

Equation Simplification

Simplifying equations is a helpful skill in solving geometric problems. In our example, the given equation \( 3(x+2)+5(y-9)-4z=0 \) needs simplification to ease finding solutions. The goal is to transform it into a clearer form without altering its fundamental meaning.
The first step involves distributive property: multiply the constants through the parentheses. This yields \( 3x + 6 + 5y - 45 - 4z = 0 \). Combining like terms gives us \( 3x + 5y - 4z - 39 = 0 \).
Why simplify? Because a clear equation makes finding particular solutions (like points on the plane) easier, and helps reduce the chance of mistakes. Simpler forms are also easier to visualize or graph which aids both in understanding and further calculation steps.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to study geometric figures using a coordinate system. By using coordinates, we can perform calculations and work through problems involving points, lines, and planes in space.
For planes, coordinate geometry involves solving equations to find points that lie on the plane. For example, to find specific points on the plane given by \( 3x + 5y - 4z - 39 = 0 \), we substitute chosen values for one or two variables and solve for the remaining one(s).
The coordinates of points we solve for, like \((0, 9, 1.5)\) and \((1, 10, 3.5)\), represent solutions that satisfy the plane's equation. This method is useful for visualizing planes, drawing them on graphs, or solving practical problems where the spatial arrangement is crucial. This is why coordinate geometry is an essential part of both math education and applications in scientific fields.