Welcome to the first of several examples
on how to solve two-step equations with fractions.
We're actually going to take a look at each equation
twice using two different methods.
One method will leave the fractions in the equation
and solve the equation like any other two-step equation,
and for the second method, which we'll show in this video,
we're actually going to clear the fractions
from the equation by multiplying everything
in the equation by the least common denominator
and then solve the resulting equation.
So because we want to clear the fractions from the equation,
we're going to solve the equation using the following steps.
Number one, which is an optional step, we're going to build
fractions that have the same least common denominator.
And then number two, we'll multiply each fraction
by the common denominator to clear the fractions
from the equation.
And then we'll solve by adding or subtracting to isolate
the variable term and then we'll multiply or divide
to solve for the variable
and finally check our solution.
So looking at our example, notice that 3x is not
in fraction form, so I'm going to go ahead and rewrite
this as 3x over one
plus 3/5
equals 3/4.
So we're going to build these fractions
and then we want to obtain the least common denominator
of one, five and four
which would be the least common multiple of five and four
which is 20.
Another way to think of this is 20 is the smallest number
that's divisible both both four, five and one.
So if have a denominator of 20 here
we have to multiply the numerator and denominator by 20.
If we want a denominator of 20 here, we'd multiply
both the numerator and denominator by four.
If we wanted to have a denominator of 20 here,
we'd multiple the numerator and denominator by five
since four times five is 20.
So now we actually have 60x over 20
plus 12 over 20
equals, this will be 15 over 20.
Now for the fractions from the equation,
we're going to multiply each fraction by 20
or both sides of the equation by 20.
So we'll multiply this fraction by 20 or 20 over one
by this fraction by 20 or 20 over one
and multiply this fraction by 20 over one.
Notice in each case, 20 over 20 simplifies to one,
eliminating the fractions from the equation.
So now we have the equation 60x
plus 12
equals 15.
Now remember in the first step we said
building these common denominators here
is an optional step which means
if we could look at the equation and recognize
the least common denominator would be 20,
we can go ahead and multiply each term in the
original equation by 20 and the results
would be the same equation here.
Let's go in to show that if we recognize the LCD was 20,
we could jump to the step where we had
3x times 20
plus 3/5 times 20
which we can write as 20 over one
equals 3/4 times 20 or 20 over one.
I want to show this because this would be equivalent
to what we did over here on the left.
And this first term would be 60x.
Here we would simplify.
Five and 20 have a common factor of five.
There's one five in five and four fives in 20.
Notice the result would be three times four or 12.
And then on the right side
four and 20 share a common factor of four.
There's one four in four and five fours in 20.
We're left with three times five or 15,
so notice that this equation here
and this equation here are the same.
Either way, we're now going to solve this two-step equation.
So the next step we will subtract 12 on both sides.
So we have 60x equals
15 minus 12 is three
and divide both sides by 60.
So x equals 3/60 which does simplify.
Three and 60 share a common factor of three.
Three divided by three is one
and 60 divided by three is 20.
So our solution is x equals 1/20.
Now let's go ahead and check this
which means if we substitute 1/20 into the original equation
it should satisfy the equation.
We'd have three times 1/20.
I'll write this as three over one times 1/20
plus 3/5
equals 3/4.
So here we have 3/20.
If the denominator is 20, we multiply 3/5 by four over four.
That would be 12/20.
Equals 3/4, is equivalent to
15/20 and multiply it by five over five.
Well 3/20 plus 12/20 is 15/20,
so our solution checks and is correct.
So if we go back and compare this solution
to the solution that we obtained when we solved
the equation by leaving the fractions in the equation,
of course the solution is the same.
And I would probably argue in this case,
it would be easier and shorter if we just left
the equation with the fractions and work with
the fractions in the equation.
But either way, that's up for your to decide.
I hope you found this helpful.