Equations (Level 1) An equation is formed by placing an equal
sign between two numerical or variable expressions. Each equation contains two sides the left
hand side and the right hand side. In algebra equations are a type of mathematical
sentence that can either be open or closed. Examples of closed sentences are the following
numerical expressions. A closed sentence is always true or always false. In the first
equation the sentence is always true, 3 plus 5 is equal to 8 is always true,
in the second equation the sentence is always false. 4 plus 4 is equal to 10 is a false statement.
Equations containing variables are called open sentences because of the fact that they
can either be true or false depending on the value of the variable. Examples of open
sentences are the following variable expressions.
the equation 5x + 1 = 11 and the equation y + 5 = 5 + y
At times the values that a variable can attain can be restricted to one or more numbers.
The set of numbers that a variable may represent is called the domain.
As a consequence, a variable in an equation can be replaced by any of the numbers in its domain.
Remember the resulting equation may be either true or false.
In algebra the way we denote a set of numbers is by enclosing them using braces { }
for example a short way to write “the set whose members are 1, 2 and 3” is as follows,
moreover this set also corresponds to the domain of a specific variable. In this
example the domain consist of the numbers 1, 2, and 3.
Ok
Say we have the following equation 6x - 1 = 17 and we are asked to solve this equation
over the domain 2, 3, 4, and 5 in other words the variable x is a set whose members are 2, 3, 4, and 5.
The way we mathematically represent the domain is as follows, this expression is read as
“x belongs to the set whose members are 2, 3, 4, and 5. It can also be read as “x
is an element of” 2, 3, 4, and 5.
The purpose of this problem is to find the solution set of the equation over a given
domain. Any value of a variable that turns an open sentence, in this case the equation;
into a true statement is a solution or root of the equation. In other words it is said
to satisfy the equation.
So the main goal of this problem is to find the values of x from the given domain that
yields a true statement. The easiest way to solve this problem is by creating a table
as follows: The first column will contain all the elements
of the given domain in this case 2, 3, 4, and 5.
The second column is where we will substitute the given elements
into the equation and evaluate them at the given number; the third column will contain
the simplified equation, and the last column will keep track if the resulting substitution
yielded a true or false statement.
Ok so let’s systematically go though each element to find the solution set.
The first element in the domain is the number 2 so we substitute that into the equation
and evaluate it at that given number this results in the statement 11 = 17 which is
false. Let's evaluate the second element in the domain
in this case its 3 and substituting it yields the statement 17 = 17 which is true.
The third element is 4 so substituting it into the equation yields the statement 23
= 17 which is false. Now the final element of the domain is 5,
substituting it into the equation yields the statement 29 = 17 which is false.
Alright our job is to find all solutions of the equation from the elements of the given
domain, looking at the final column we see that the only element that yielded a true
statement was the number 3, therefore the solution set is the number 3.
The way you mathematically represent the answer is as follows, we use 3 dots arranged in the
form of a triangle this symbol is read “therefore” and the solution or roots of the equation
is enclosed in braces like this,
We conclude that the solution set is equal to 3.
Be aware that some equations may have more than one solution, and some equations may have
no solutions at all. An example of an equation that has more than one solution is the sentence
x + 2 = 2 + x this equation is true no matter what number is substituted for x. In the end
the solution set to an equation depends on the elements of the domain that are given.
Alright. In our next video will go over more examples in finding the solutions sets of equations
over a given domain.