Beginning Activity Beginning Activity 1: An Equation with Two Variables
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, \(\left( 3, 2 \right)\) is a solution of the equation \(2x + 3y = 12\text{,}\) but \(\left( 2, 3 \right)\) is not a solution of this equation.
1.
List six different elements of the truth set (often called the solution set) of the open sentence with two variables \(2x + 3y = 12\text{.}\)
2.
From previous mathematics courses, we know that the graph of the equation \(2x + 3y = 12\) is a straight line. Sketch the graph of the equation \(2x + 3y = 12\) in the \(xy\)-coordinate plane. What does the graph of the equation \(2x + 3y = 12\) show?
3.
Write a description of the solution set \(S\) of the equation \(2x + 3y = 12\) using set builder notation.