Question

Regarding \(4 x-2 y=3\) as the equation \(4 x-2 y+0 z=3\) in three variables, find all solutions in parametric form.

Short answer

All solutions are \(x=s\), \(y=2s-\frac{3}{2}\), \(z=t\) where \(s,t\in\mathbb{R}\).

Step-by-step solution

Rewrite the Equation

Start by recognizing that the equation \(4x - 2y + 0z = 3\) represents a plane in three-dimensional space. By acknowledging the presence of a zero coefficient for \(z\), we can treat \(z\) as a free variable.

Solve for One Variable

We choose to solve for \(y\) in terms of \(x\) and \(z\), keeping \(z\) as a free parameter. Rearrange the equation:\[ 4x - 2y = 3 \]to solve for \(y\):\[ -2y = 3 - 4x \]\[ y = 2x - \frac{3}{2} \].

Parameterize the Free Variable

Since \(z\) is not constrained by the equation, we declare it as a parameter: let \(z = t\), where \(t\) is any real number. This parameterization guides us in expressing \(x\) in terms of \(t\) by another free parameter.

Express General Solution

The variable \(x\) can also be free. Thus, let \(x = s\), where \(s\) is any real number. Substitute \(x = s\) into the expression for \(y\):\[ y = 2s - \frac{3}{2} \]. The parametric solution in terms of \(s\) and \(t\) becomes:\[ \begin{align*}x &= s, \y &= 2s - \frac{3}{2}, \z &= t. \end{align*} \]

Verify the Parametric Solution

Verify the parametric expression satisfies the original equation. Substituting into \(4x - 2y + 0z = 3\):\[ 4(s) - 2(2s - \frac{3}{2}) = 3 \]\[ 4s - 4s + 3 = 3 \]\[ 3 = 3 \]This verification shows that the parametric expression is indeed a solution.

Key concepts

Three-Dimensional Space

Three-dimensional space is like a mathematical playground where we explore objects that have three directions for movement: left-right (x-axis), forward-backward (y-axis), and up-down (z-axis). This space is often visualized with a 3D graph or coordinate system, where every point in space is represented by an ordered triplet \(x, y, z\). Here, each variable corresponds to a dimension, making three-dimensional space a convenient setting for solving complex problems involving multiple dimensions.
When dealing with planes, lines, and other geometric figures, three-dimensional space offers a richer environment. In such space, an equation like \(4x - 2y + 0z = 3\) describes a plane. This means we're looking at a flat surface that cuts through the space. Unlike a line which is one-dimensional, a plane extends in two dimensions within the three-dimensional setting. Our problem revolves around such a plane, where we explore how variables interact on this flat surface.

Plane Equation

A plane equation like \(4x - 2y + 0z = 3\) is a mathematical statement that defines a flat surface within three-dimensional space. This particular equation shows a relationship between x, y, and z coordinates. In the equation, each term corresponds to one of the variables. The coefficient of a variable indicates how it contributes to forming the plane. The constant term, here 3, determines the plane's position relative to the origin.

The equation can be written in a general form, \(Ax + By + Cz = D\), where A, B, and C are coefficients and D is a constant. If a coefficient, like C in \(4x - 2y + 0z = 3\), is zero, it implies the plane is parallel to the corresponding axis. Since z has a zero coefficient, changes in z don't affect this plane, making the plane infinitely wide in the z-direction. By setting up the equation correctly, we find that two directions (x and y) form the plane while one (z) is free, meaning the plane stretches along that axis.

Free Variable

A free variable arises in equations when its coefficient is zero, suggesting it doesn't affect the equation's outcome. In our equation, \(4x - 2y + 0z = 3\), z is a free variable because its coefficient is 0. This means z can be any value, and the equation will still hold true. Free variables are incredibly helpful in parametric equations, helping us explore all possible solutions to an equation without being restricted by specific values for every variable.
When treating z as a free variable, it becomes a parameter, often denoted as t. This freedom lets us express other variables like x and y in terms of t. Such parameterization helps define a family of solutions that illustrate how x and y relate to different z values. Using parameters streamlines our work and reveals the full set of solutions for an equation lying on a specified plane.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear equations, often used to solve equations in a structured manner. It forms the backbone of problem-solving in three-dimensional space. Our equation \(4x - 2y + 0z = 3\) is linear since it meets the requirements for a straight line (or plane, in this context): each variable is to the power of 1, and no variables multiply each other.

Applying linear algebra to this problem, we see how the methods of parameterizing variables and handling them in an organized way come in handy. By recognizing z as a free variable and letting x also be represented as a parameter (s), we use linear algebra to express y through operations on these parameters. This systematic approach simplifies finding solutions, allowing us to explore every possible point on the plane, which is an advantage brought by the principles of linear algebra.