>> In order to solve the given linear equation in one variable, our ultimate goal is
to isolate the variable on one side of the equation,
but the first thing you'll probably notice about this equation is that it contains decimals.
Looking at our guidelines for solving multistep linear equations,
the first step is to clear any fractions or decimals.
So, for solving this equation by hand, it will be helpful to eliminate or clear the decimals
by multiplying by a power of ten based upon the number of decimal places.
Remember, the short cut for multiplying by ten is
to move the decimal point to the right one place.
The short cut for multiplying by 100 is to move the decimal point two places right and so on.
Since we have one decimal place here, one decimal place here, but two decimal places here,
if we want to clear the decimals, we'll have to multiply every term by 100.
So, let's go ahead and do that.
So, again multiplying by 100 does have the affect
of moving the decimal point to the right two places.
So, for this part, 100 times 2X would be 200X.
100 times 18.2, if you move the decimal point to the right two places,
we do have to add a zero, and we would have 1,820.
Moving the decimal point to the right twice here,
we would have 280X, and here we would have 148.
So, because this one decimal had two decimal places, we had to multiply everything by 100.
Now, looking at the equivalent equation,
notice how we do have X terms on both sides of the equation.
If our goal is to isolate X, we do have to have it on one side of the equation.
So, to get the variable terms on one side, we either have to subtract 200X on both sides
to eliminate this variable term or we have to subtract 280 X
on both sides to eliminate this term.
Notice if we subtract 200X on both sides, we would have a positive X term on the right side,
and if we subtracted 280X on both sides, we'd have a negative X term on the left side.
It doesn't matter which method we use.
But since most people prefer to have the variable term on the left side,
I'm going to go ahead and subtract 280X on both sides.
Notice here we do have zero now, and now let's go ahead and simplify here.
We're going to have negative 80X plus 1,820 equals 148.
Now, we have our basic two step equation.
We want to undo this multiplication here as well as this addition here.
So, we'll undo the addition first.
We'll subtract 1,820 on both sides.
This would be zero, so we have negative 80X equals this difference here.
If we're doing this by hand,
remember subtracting 1,820 is equivalent to adding negative 1,820.
So, this is equivalent to 148 plus negative 1,820.
So, when adding integers with different signs, we actually subtract throughout two values
and keep this under the number with a larger absolute value.
Another way to think of this is we can see the negative is going to outweigh the positive,
so this sum which is the same as it's different is going to be negative,
and now we'll just subtract their absolute values,
and we always subtract the smaller absolute value from the larger absolute value.
So, we'll have positive 1,820 as the absolute value of this negative number,
and then we'll subtract the absolute value of 148, which is 148.
We have to borrow.
Ten minus 8 is 2.
Borrow. We have 11 minus 4.
That's 7. And here we have 7 minus 1.
That's 6, and 1.
But remember, we already determined that this sum,
which is the same when it's different is going to be negative.
So, we have negative 1,672.
And then since negative 80X means negative 80 times X, we'll divide both sides by negative 80.
Here we have 1X or just X and now when you determine this quotient,
we know a negative divided by a negative is positive,
and since the equation contained decimals, we're going to go ahead
and give this solution as a decimal.
So, we'll perform this division.
We'll have 1,672 divided by 80.
So, we need to figure out how many 80's there are and 167.
That's going to be 2.
Two times 80 is 160.
Here we have a difference of 70.
Bring down the 2.
There are zero 80's in 70.
So, we put a zero here.
Zero times 80 is 0.
Subtract. That's 72.
And now we need to add a decimal.
Decimal point would be here, which will move up into the quotient, and now we can add a digit
of zero in the tens place value and bring it down.
And this is good news because there are nine 80's in 720.
So, put a 9 here.
Nine times 80 is 720.
And we subtract.
And the difference is zero.
So, this decimal terminates, and our solution is 20.9.
Remember, we know it's positive because we have a negative divided by a negative.
And that'll do it for this example.
I hope you found this helpful.