Solve the following equations.
Write each answer as an integer, proper fraction
or mixed number in simplest form.
Looking at the first equation we have the quantity
x minus six divided by eight equals 11.
The first step here will be to clear the fraction
from the equation by multiplying both sides of
the equation by the denominator of eight.
This gives us the quantity x minus six divided by eight,
and then times eight equals 11 times eight.
And now, looking at the left side of the equation,
if it's helpful, we can write eight as a fraction
with a denominator of one.
Notice before multiplying, we can see we have
a common factor of eight between
the numerator and denominator.
There's one eight and eight here, as well as here,
or we can think eight divided by eight is equal to one.
Notice now the left side is just the quantity x minus six,
which is why we multiplied both sides of
the equation by eight.
On the right side we have 11 times eight, which is 88.
And now we have a basic one step equation to solve for x.
We need to undo the subtraction by adding six to
both sides of the equation.
Simplifying, negative six plus six is zero,
which is why we added six to both sides.
We now have x equals 88 plus six, which is 94.
The solution is x equals 94.
For the second equation, we have the quantity
8x plus five divided by seven equals nine.
So again, the first step here will be to
clear the fraction from the equation by multiplying
both sides of the equation by the denominator of seven.
This gives us the quantity 8x plus five
divided by seven times seven,
or if we want times seven over one,
equals on the right side nine times seven.
And now, back to the left side
we can see we have a common factor of seven
between the numerator and denominator.
There's one seven and seven here, as well as here.
Again, we can think seven divided by seven is equal to one,
so now on the left side we just have the quantity
8x plus five times one, which is 8x plus five.
On the right side, we have nine times seven, which is 63.
Now we have a two step equation.
The next step is to isolate 8x by undoing the addition,
which means the next step is to subtract five on both sides.
Simplifying, five minus five is zero,
giving us 8x on the left.
On the right, 63 minus five is 58.
From here, since 8x means eight times x,
to solve for x we undo the multiplication
by dividing both sides by eight.
Simplifying, eight divided by eight simplifies to one.
One times x is x.
We have x equals 58/8, which can be simplified, though.
The greatest common factor between 58 and eight is two.
To simplify the improper fraction
we divide the numerator and denominator by two.
58 divided by two is 29
and eight divided by two is four.
So, we do have a simplified improper fraction here,
but we're asked to express the answer as an integer,
proper fraction, or a mixed number.
And since we have an improper fraction
because of the directions,
we'll have to convert this to a mixed number.
To convert an improper fraction to a mixed number
we divide the numerator by the denominator.
So, we have 29 divided by four.
There are seven fours in 29.
Seven times four is 28.
We subtract, and the remainder is one,
which means we can express the quotient as seven and 1/4.
Notice how we form this fraction
by placing the remainder over the divisor.
So, our solution is also x equals seven and 1/4.
I hope you found this helpful.