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Beginning Activity Beginning Activity 2: The Solution Set of an Equation with Two Variables
A4US

In Section 2.3, we introduced the concept of the truth set of an open sentence with one variable. This was defined to be the set of all elements in the universal set that can be substituted for the variable to make the open sentence a true proposition. Assume that \(x\) and \(y\) represent real numbers. Then the equation

\begin{equation*} 4x^2 + y^2 = 16 \end{equation*}

is an open sentence with two variables. An element of the truth set of this open sentence (also called a solution of the equation) is an ordered pair \(\left( {a, b} \right)\) of real numbers so that when \(a\) is substituted for \(x\) and \(b\) is substituted for \(y\text{,}\) the predicate becomes a true statement (a true equation in this case). We can use set builder notation to describe the truth set \(S\) of this equation with two variables as follows:

\begin{equation*} S = \left\{ (x, y) \in \R \times \R \mid 4x^2 + y^2 = 16 \right\}\!\text{.} \end{equation*}

When a set is a truth set of an open sentence that is an equation, we also call the set the solution set of the equation.

1.

List four different elements of the set \(S\text{.}\)

2.

The graph of the equation \(4x^2 + y^2 = 16\) in the \(xy\)-coordinate plane is an ellipse. Draw the graph and explain why this graph is a representation of the truth set (solution set) of the equation \(4x^2 + y^2 = 16\text{.}\)

3.

Describe each of the following sets as an interval of real numbers:

(a)

\(A = \left\{ x \in \R \mid \text{ there exists a } y \in \R \text{ such that } 4x^2 + y^2 = 16 \right\}\text{.}\)

(b)

\(B = \left\{ y \in \R \mid \text{ there exists an } x \in \R \text{ such that } 4x^2 + y^2 = 16 \right\}\text{.}\)