Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

Beginning Activity Beginning Activity 1: 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 statement.

In previous mathematics courses, we have also had experience with open sentences with two variables. For example, if we assume that $x$ and $y$ represent real numbers, then the 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 open sentence becomes a true statement (a true equation in this case). For example, we see that the ordered pair $(6,0)$ is in the truth set for this open sentence since

is a true statement. On the other hand, the ordered pair $(4,1)$ is not in the truth set for this open sentence since

is a false statement.

Important Note: The order of the two numbers in the ordered pair is very important. We are using the convention that the first number is to be substituted for $x$ and the second number is to be substituted for $y\text{.}$ With this convention, $(3,2)$ is a solution of the equation $2x+3y=12\text{,}$ but $(2,3)$ is not a solution of this equation.