Many algebra textbooks, including
ours have problems where you are supposed
to translate sentences similar to
the following one into equations.
Three less than the product of four
and a number is the product of four
and the sum of the number and seven.
Here is a method that I find helps
students keep from getting tied in
knots with this.
Find the word 'is' in the sentence
and change it into '='.
Three less than the product of four
and number = the product of
four and the sum of the number and
seven.
The equals will divide the sentence
into two verbal expressions than can
be translated into variable expressions
by the methods of my article Translating Verbal Expressions to Variable
Expressions.
In this case for the first expression
it would go like this.
three less than the product of
four and a number
The first thing that would make sense
alone here is 'the product of four
and a number', so put parentheses
around it.
three less than (the product of
four and a number)
'the product of four and a number'
can be translated to 4x, so this is
three less than (4x).
If I want to compute the number than
is three less than a number, then
I subtract 3 from the number, so this
becomes
4x-3
I can drop the parentheses around
the 4x here, because multiplication
comes before subtraction in the order
of operations agreement.
For the second expression we have
the product of four and the sum
of the number and seven
'the sum of the number and seven'
is an expression that can stand alone,
so put parentheses around it.
the product of four and (the sum
of the number and seven)
It can be translated to x+7, so this
becomes
the product of four and (x+7)
This then becomes
4(x+7).
Then putting the whole sentence together,
it says
4x-3=4(x+7).
See Linear Equations to find out how to solve it.
