Video Transcript
If the arithmetic mean between 𝑎
and 𝑏 is nine and the arithmetic mean between seven 𝑎 and five 𝑏 is 15, then 𝑏
minus 𝑎 equals blank.
Let’s start out by defining what we
mean by arithmetic mean. The arithmetic mean is the sum of
the values divided by the number of values. And that means we can set up an
equation for the arithmetic mean of 𝑎 and 𝑏, which would be 𝑎 plus 𝑏 divided by
two. We need the sum of the values of 𝑎
and 𝑏, and there are two values, so we divide that by two. But we already know that this mean
is equal to nine. We can set up a second equation for
the arithmetic mean of seven 𝑎 and five 𝑏. This would be seven 𝑎 plus five 𝑏
divided by two.
Again, we’re only dealing with two
terms, and that means we need to divide their sum by two. And we’ve been told that this is
equal to 15. At this point, we’ll want to
rearrange the equations to see if we can solve simultaneously. On the left, if we multiply both
sides of the equation by two, we get 𝑎 plus 𝑏 equals 18. And if we do the same thing with
our second equation, we’ll get seven 𝑎 plus five 𝑏 equals 30. Now that we have two equations with
variables 𝑎 and 𝑏, we could solve equation one for one of our variables and use
substitution.
If we wanted to solve
simultaneously, we see that in our second equation our 𝑏-variable is five 𝑏. So if we take our first equation
and multiply through by negative five, we get an equivalent equation that says
negative five 𝑎 minus five 𝑏 equals negative 90. And now we can combine both of
these equations and eliminate that 𝑏-variable. Since five 𝑏 minus five 𝑏 equals
zero, seven 𝑎 minus five 𝑎 equals two 𝑎, and 30 minus 90 equals negative 60. If two 𝑎 equals negative 60, then
𝑎 equals negative 30. And we know from our first equation
that 𝑎 plus 𝑏 must equal 18. So we plug in negative 30 for 𝑎
and add 30 to both sides of the equation to give us 𝑏 equals 48.
We haven’t finished solving here
since we’re trying to solve for 𝑏 minus 𝑎, which will be 48 minus negative 30. Make sure you recognize that 𝑎 is
a negative value. We’re subtracting a negative. Negative 48 minus negative 30 is
equal to 48 plus 30, which is 78. Therefore, 𝑏 minus 𝑎 equals
78.
