The solution to the equation
is the value of the variable that satisfies
the equation or makes the equation true.
In order to solve the equation, we need
to isolate the variable on one side of the equation.
Also notice this equation contains fractions.
So we need to decide whether we want to work
with the fractions to isolate the variable
or to first clear the fractions from the equation
and then isolate the variable.
In this video, we will practice clearing
the fractions from the equation by multiplying
both sides of the equation by the least common denominator.
Notice how we have a denominator of nine
and three, and since the least common multiple
of nine and three is nine, we will clear
the fractions from the equation by multiplying
both sides of the equation by nine.
However, if we have a hard time determining
the least common denominator, we could multiply
both sides of the equation by nine times three, which is 27.
That would just require more simplifying later.
So let's go ahead and multiply both sides
of the equation by nine.
And now let's show each product.
We have nine times 2/9 X minus nine times one
equals nine times 2/3 X minus nine times two.
For the products involving fractions, let's write
nine as a fraction with a denominator of one.
And now before multiplying, we will simplify
out the common factors between the numerators
and denominators.
Notice for the first product the numerator
and denominator share a common factor of nine.
Simplifying, nine divided by nine simplifies
to one in the numerator and denominator.
Now we just have one times two X, which is two X,
minus nine times one, which is nine, equals on the right.
The numerator and denominator share a common
factor of three.
Simplifying, a three divided by three simplifies
to one and nine divided by three simplifies to three.
We now we have three times two X, which is six X,
and then minus nine times two is minus 18.
Now we have an equivalent equation
without fractions that we can solve.
The first step is add or subtract to isolate
the variable terms on one side of the equation.
Notice right now we have two X on the left
and six X on the right, which means we can either
subtract six X on both sides or subtract two X
on both sides.
Let's go ahead and subtract six X on both sides.
Simplifying, two X minus six X is negative four X.
The left side simplifies to negative four X
minus nine equals on the right.
Six X minus six X is zero, which is why
we subtracted six X on both sides.
The right side simplifies to negative 18.
So adding or subtracting to isolate
the variable term, to undo the minus nine
we add nine to both sides.
Simplifying, negative nine plus nine is zero.
The left side simplifies to negative four X,
which is equal to negative 18 plus nine,
which is negative nine.
The last step is to multiply or divide in order
to solve for X or isolate the variable.
Negative four X means negative four times X.
To undo the multiplication and solve for X
we divide both sides by negative four.
Simplifying, negative four divided by itself
simplifies to one.
One times X is X.
We have X equals negative nine divided
by negative four is equal to the positive 9/4,
which doesn't simplify and therefore X equals 9/4
is the solution.
This means if we substitute 9/4 for X
into the original equation it will satisfy
the equation or make the equation true.
And let's go ahead and check this.
Performing the substitution for our check
we have 2/9 times X, which is 9/4, minus one
equals on the right 2/3 times 9/4 minus two.
And now let's evaluate both expressions on the calculator.
Notice the expression on the left
and on the right are both equal to negative 0.5.
Well, if we want negative 1/2, our solution is correct.
Again, the expression on the left and the right
are both equal to negative 1/2, which is true,
verifying the solution is correct.
I hope you found this helpful.