Question

Describe the set of all points at which all three planes \(x+3 z=3, y+4 z=6,\) and \(x+y+6 z=9\) intersect.

Short answer

Question: Determine the set of points where all three planes given by the equations \(x + 3z = 3\), \(y + 4z = 6\), and \(x + y + 6z = 9\) intersect. Answer: The set of points where all three planes intersect can be described by the parametric equations \(x = 3 - 3z\) and \(y = 6 - 4z\).

Step-by-step solution

Write the system of linear equations

From the given planes, we can write the system of linear equations as: 1. \( x + 3z = 3 \) 2. \( y + 4z = 6 \) 3. \( x + y + 6z = 9 \)

Solve for one variable in terms of another variable in one of the equations

Let's solve for x in equation 1. \(x = 3 - 3z\)

Substitute the value of x in the third equation

Replace x in equation 3 with the expression found in Step 2. \((3 - 3z) + y + 6z = 9\)

Simplify the equation and solve for y

Simplify the equation obtained in Step 3: \(3z + y = 6\) Since this equation is the same as equation 2, we can say that equations 2 and 3 are dependent. Now we have to check if there is a common solution (or infinite solutions) for the system of equations.

Substitute the value of x and y in terms of z into the first equation

Use the expression for x we found in step 2 and for y which we can re-write from equation 2: \(x = 3 - 3z\) \(y = 6 - 4z\) Substitute these expressions in equation 1: \((3-3z) + 3z = 3\)

Analyze the result

From Step 5, we have: \(3=3\) This equation is true for all values of z. Therefore, the system of equations has infinitely many solutions and the three planes intersect along a common straight line, which is parameterized by the variable z.

Write the parametric equation for the intersection line

Using the expressions for x and y in terms of z from Steps 2 and 4, the parametric equation of the intersection line can be written as: \(x = 3 - 3z\) \(y = 6 - 4z\) Thus, the set of all points where all three planes intersect can be described by the above parametric equations.

Key concepts

Intersection of Planes

When you deal with systems of linear equations representing planes in three-dimensional space, the 'intersection of planes' is a crucial concept. Each plane can be visualized as an infinite flat surface extending across three dimensions. When there are three such planes, they can intersect in different ways:

• At a single point, if all three planes meet exactly at one common point.
• Along a line, if they share a common line of intersection. This means the planes intersect in such a way that they all lie along the same straight path.
• There might also be no common intersection if the planes are parallel or arranged such that they do not meet at any point.

In the given exercise, the equations represent three planes: 1. Plane 1: \(x + 3z = 3\)2. Plane 2: \(y + 4z = 6\)3. Plane 3: \(x + y + 6z = 9\)The solution process showed that these three planes intersect along a common line. That's because the system of equations simplifies to a single statement, \(3=3\), indicating that the equations describe lines that lie together. Understanding where and how planes intersect can help in solving many practical and theoretical problems in science, engineering, and mathematics.

Parametric Equations

Understanding parametric equations is essential for describing the set of points that form lines or curves in space. A parametric equation uses one or more parameters to describe a curve or line. In this instance, the intersection line of the planes is parameterized using the variable \(z\).Let's examine what this means for our problem:

• Solutions for \(x\) and \(y\) are expressed in terms of \(z\):
• \(x = 3 - 3z\)
• \(y = 6 - 4z\)
• \(z = z\) (as \(z\) itself is the parameter)

These equations allow us to describe any point along the line of intersection by simply choosing a value for \(z\). Different values of \(z\) will give different points on the intersection line. Parametric equations are particularly useful in physics and engineering for modeling motion among other applications because they can easily represent points in motion or transformation over time.

Linear Algebra

Linear Algebra provides the fundamental tools to deal with vector spaces and linear equations. It helps us understand systems of linear equations, like the one described by our three planes. In linear algebra, we often work with matrices and vectors to represent and solve these systems. For example, each of our equations can be thought of in matrix form to see how they relate as a system. Understanding how to manipulate these matrices or recognize dependent equations is critical:

• If two or more equations in the system are dependent, it means they represent the same plane or intersect in ways that reveal parallelism or a line intersection.
• Operators like Gaussian elimination are used to simplify such systems by finding solutions or showing how planes could interact.

The solution to our specific problem revealed that the system had infinitely many solutions, characterized by a common line of intersection for all three planes. Linear algebra provides the methodologies to explore these concepts systematically, aiding in everything from designing complex networks to understanding natural phenomena.