Equations (Level 2) Alright let's work on the following examples,
Solve the equation x + 5 = 11 if x is an element of 0, 2, 4 and 6.
Like in the previous video let's set up a table as follows, the first column will contain
the elements of the domain in this case 0, 2, 4 and 6, the second column is where we substitute
the variable of the equation with its respective element. The third column will contain the
simplified equation after substituting the elements into the equation and lastly the
final column will keep track if the statement obtained is either true or false.
So substituting the first element of the domain into the equation and simplifying yields the
statement 5 = 11 which is absolutely false. Next we substitute the second element of the
domain in this case 2 into the equation which simplifies to the statement 7 = 11 which is
false. Moving along we substitute the third element of the domain the number 4 into the
equation and simplifying the expression yields the statement 9 = 11 which is also false. Lastly,
we substitute the final expression of the domain in this case the number 6 into the
equation and then we simply the expression which reduces to statement 11 = 11 which is
a true statement. Looking at the results we see that only one element in the domain yielded
a true statement therefore the solution set is 6.
Alright let’s try the next example. Solve the equation x times the quantity 3
minus x is equal to 0 if x is an element of 0, 1, 2 and 3.
Alright just like the previous problem let’s set up a table and start systematically
substituting the elements of the domain into the equation. The first element is zero so
we substitute that into the equation which yields the statement 0 = 0 which is a true
statement. The second element of the domain is 1 so substituting that into the equation
yields the statement 2 = 0 which is false. The third element is 2 so substituting that into
the equation yields the statement 2 = 0 which is also false. Lastly, the element 3 is substituted
into the equation which yields the statement 0 = 0 which is true. Looking at the results
we see that two elements of the domain yielded a true statement, therefore the solution set
is 0 and 3.
Notice that in this example the equation had more than one solution or root that satisfied
the equation in other words the element lead to a true statement.
Alright let’s try the next example. Solve the equation z + 5 = z + 5 if z is an
element of 0, 1 and 2.
Once again we set up a table as follows and systematically go through each element of
the domain, so substituting the first element we have the statement 5 = 5 which is a true
statement. The second element is the number 1 and substituting that into the equation
yields the statement 6 = 6 which is also a true statement. Lastly the third element of
the domain is the number 2, substituting it into the equation yields the statement 7 = 7
which is also a true statement. Notice that every single element in the domain yielded
a true statement therefore the solution set is 0, 1 and 2.
Let’s try the next example. Solve the equation 2x = x + 7 if x is an element
of 0 and 1.
Notice that we only have 2 elements in the domain. Like always let’s create a table
and substitute each element of the domain into the equation. Substituting the first
element into the equation and simplifying we have the statement 0 = 7 which is false.
The last element in the domain is 1 and after substituting it into the equation we have
the statement 2 = 8 which is also false. Looking at the results we see that there isn't a single
element in the domain that yielded a true statement. This means that the equation has
no solution over the given domain. In this example there wasn't an element in the given
domain that yielded a true statement meaning there wasn't a solution or root that satisfied
the equation.
Recall that the solution set is dependent on the domain given for example if we work
with the same equation but with a completely different domain say the domain that contains
the elements 6 and 7.
We would obtain a completely different answer, so let's try it out substituting the first element into the
equation yields the statement 12 = 13 which is a false statement, the last
element yields the statement 14 = 14 which is a true statement. Looking at the results
we see that only one element yielded a true statement therefore the solution set is equal to 7. Notice
that the solution to the equation is dependent on the domain for which we are solving the
equation, this is an important concept that many students from higher level mathematics
such as calculus and differential equations tend to forget.
Alright, in our next video we will learn how to translate written words into numerical
and variables expressions.