We're going to solve the equation
4/5ths x minus five halves
equals negative 9/10ths x minus seven.
The solution is a value of x that satisfies the equation
which means the solution is a value of x
that makes the expression on the left
equal to the expression on the right.
And because the equation has fractions,
we need to decide whether we want to solve the equation
with the fractions in it or clear the fractions
by multiplying both sides of the equation
by the least common denominator,
or at least a common denominator.
In this video, we'll be solving the equation
by first clearing the fractions from the equation.
Let's see if we can identify the least common denominator.
The least common denominator is the least common multiple
of five, two, and 10,
which is also the smallest number
best divisible by five, two, and 10.
And because the least common multiple of five, two, and 10
is 10, we'll clear the fractions from the equation
by multiplying both sides of the equation by 10.
We can clear the fractions by multiplying
by any common denominator,
but if we multiply by the least common denominator,
there'll be less simplifying later.
So let's multiply the left side of the equation by 10
and the right side by 10.
And let's go ahead and show which product.
So we'd have 10 times 4/5ths x
minus 10 times five halves
equals 10 times negative 9/10ths x
minus 10 times seven.
And now for the products involving a fraction,
let's write 10 as a fraction with the denominator of one.
And that will determine each product.
Before multiplying though, we'll simplify.
Notice the numerator, denominator share a common factor
of five here.
There's one five in five
and two fives in 10.
The product is now just two times four x
which equals eight x
minus, here we have a common factor of two,
there is one two in two and five twos in 10.
The product is now just five times five which equals 25,
so we have minus 25.
On the right side, we have a common factor of 10 here.
There is one 10 in 10 and one 10 in 10.
The product is now one times negative nine x,
which is negative nine x, and then we have
minus 10 times seven which equals 70.
Notice now the equation no longer contains fractions.
In order to solve for x, we need to isolate x
on one side of the equation.
And notice right now, we have an x term on the left
and an x term on the right.
So for the next step, we want all the x terms
to be on one side,
and therefore, we need to undo or eliminate
the negative nine x on the right
or undo or eliminate the eight x on the left.
Let's undo or eliminate the negative nine x
on the right side by adding nine x to both sides
of the equation.
Notice adding nine x will undo the negative nine x
on the right side.
Whatever operation we performed to one side,
we must also perform to the other.
So add nine x to both sides of the equation
and simplify.
Eight x plus nine x is 17 x.
So we have 17 x minus 25 equals
negative nine x plus nine x is zero,
that's why we added nine x to both sides.
So the right side is now just negative 70.
So now we have the equation 17 x
minus 25 equals negative 70.
The next step is to undo the subtraction
so we can isolate the variable term 17 x.
To undo minus 25,
we want to perform the opposite or inverse operation,
and therefore we'll add 25 to both sides of the equation.
Adding 25 will undo the minus 25.
And we simplify both sides of the equation.
Again, adding 25 undoes minus 25,
or negative 25 plus 25 is zero.
So now we have the equation 17 x equals.
On the right side we have negative 70 plus 25
which equals negative 45.
Finally, our last step is to solve for the variable x,
and because 17 x means 17 times x,
we want to undo the multiplication involving 17,
and therefore the next step is to divide both sides
by 17.
Dividing by 17 undoes multiplying by 17,
or 17 divided by 17 equals one,
and one times x is x equals,
on the right side we have the fraction
negative 45/17ths.
And because the only common factor between 45 and 17 is one,
the fraction is in simplest form,
and we now have our solution.
The solution is x equals negative 45/17ths.
Before we go though, let's verify our solution
by substituting negative 45/17ths for x
in the original equation and make sure it does
satisfy the equation.
So performing the substitution, we'd have 4/5ths
times negative 45/17ths
minus five halves
equals negative 9/10ths times negative 45/17ths
minus seven.
Now let's use a calculator to evaluate the left side
and the right side of the equation.
On the left side, we have 4/5ths
times negative 45/17ths
minus five halves.
Enter.
Let's convert this to a fraction by pressing math,
enter, enter.
So the left side is equal to a negative 157/34ths.
Let's make sure the right side is also
negative 157/34ths.
So on the right side we have negative 9/10ths
times negative 45/17ths
and then minus seven.
Enter.
Math, enter, enter converts the decimal to a fraction,
and notice how we have the same value on the right side
when x equals negative 45/17ths,
verifying our solution is correct.
On the left, we have negative 157/34ths,
and we have the same value on the right
which is our check that our solution is correct.
I hope you found this helpful.