An equation is a mathematical
statement of equality written by placing an equal sign, =, between
two expressions.
For example, 2x + b is one
expression and 3y is another expression, a statement of equality
would be:
2x + b = 3y
A formula is an equation that
relates measurements.
For example:
A = lw The area A of a
rectangle is the product of its length l and width w.
P = 2l + 2w The perimeter P of a
rectangle is twice its length 2l and twice its width 2r.
V = (4/3)∏
r3 The volume V of a sphere is 4/3 of the product of its
radius r times the constant ∏
( ≈ 3.14159 ).
Writing
a formula or an equation requires the ability to detect the
operations between quantities and to assign appropriate variables to
represent these quantities.
Examples:
1.
Bill is t cm tall. Larry is 5 cm shorter than Bill. How tall
is Bill?
Assign
L as the height of Larry. "Shorter" implies subtraction
(from Larry's point of view) and, "is" is always "=,"
so we we can write the formula
L
= t - 5 cm
2.
You buy (n - 2) bars of candy at $1.24 each. What is the total
cost?
Assign
T for the total cost and we get
T = 1.24 * (n - 1)
3.
You travel for time ( t - 15 ) hours at 60 miles per hour. The
distance traveled is?
Assign
D for distance. Speed multiplied by time is distance traveled.
D
= 60 ( t - 15 )
4.
The length of a rectangle is 20 less than half its width. The
length is? The perimeter is?
Assign
L for length and W for width. Directly read the statement,
length
L, "is" = , - 20 + 1/2 W. So we get
L
= 1/2 W - 20
"less
than" : subtract from what follows
"more
than" : add from what follows
Assign P for perimeter, then we
have
P
= 2 L + 2 W
5.
A basketball team played g games and won 3 times as many games at
it lost and tied two. Write a formula for number of games lost.
Assign
l for games lost. So 3 times lost games is 3 l, number of games
won. So, we get
l
+ 3 l + 2 = g
combine
the ls
4
l + 2 = g
subtract
the 2
4 l = g - 2
divide
by 4
l = (g - 2) / 4
6.
A policeman drove from home to his precinct and back in 85 minutes.
The return trip took 5 minutes longer. Find the time it took to
drive to the precinct.
Let
t be the time to the precinct. The round trip took 85 minutes. So
time
to the precinct t plus the return time t + 5 is 85. We can
write:
t
+ (t + 5) = 85
combine
the ts
2
t + 5 = 85
subtract
the 5
2
t = 80
divide
by 2
t = 40 mins
For
example: (with full explanation)
Rick
paid $5 less than Josh for their lunch which totaled $12.38.
How
much did each pay for lunch?
1)
first assign appropriate variables for each quantity being related
and identify with words what each represents.
Now
rereading the statement, a hint to what these quantities are lies in
the final question: how much did each pay.
Ok,
now you might assign the variables R for Rick's amount and J for
Josh's amount. This seem natural, so let's give it a try.
R
is the amount Rick paid.
J
is the amount Josh paid.
2)
From the statement we know Rick paid $5 less than Josh.
This
means that if Josh paid $7 then $5 less would be $2, the amount Rick
paid. We have subtraction here, the word "less" always
means subtraction.
So
from this we know that the amount Rick paid R must be the amount
Josh paid J minus $5. Writing this out we have R = J - 5. This
is well and good but does it answer the question? No. But we're ok.
It's time to look at the whole statement now. What they paid
totaled $12.38. Total always means the sum of two or more
quantities and implies equals. We're talking about the total of what
they paid. Writing this statement into an equation we have:
R + J = 12.38
Now
we're getting somewhere. We still haven't answered the question. We
have two variables in this equation, we need one. Well we know that
Rick paid $5 less than J, R = J - 5. Now let's stop for a minute.
An equation states that the expression on the left of the equal sign
is identical to the expression on the right of the equal sign. In
other words, they can be used interchangeably. This fact is MOST
important. :)
So,
in any (related) equation wherever we see an R we can replace it with
J - 5.
In
our case we can write J - 5 for the R in the equation R + J =
12.38.
Doing
that replacement we get
J - 5 + J = 12.38.
This
replacement is called "direct substitution" and
parenthesis are usually used to show this. In our case it would be
( J - 5 ) + J = $12.38
Now
we have an equation with a single variable, J. If we can find what
Josh paid, the $5 less than that would be the amount Rick paid.
We
can add like variables. In this case J.
2 J - 5 = 12.38
What
we need to do is get the variable J all by itself onto one side of
the equation and every thing else over on the other side.
To
do this we can add, subtract, multiply or divide by any
quantity/number, but we MUST operate on BOTH sides of
the equation or we lose the equality.
The
number 0 does not count when dividing or multiplying; in the first
case dividing by 0 has no meaning and in the second case, multiplying
by zero eliminates the problem altogether! :)
So
with this said, let's get rid of the 5 on the left hand side, by
adding 5 from both sides (always verbally say "both sides,"
so you do not forget)
2
J - 5 + 5 = 12.38 + 5
doing
the addition we get
2
J + 0 = 17.38
or
just
2 J = 17.38
This
equation is telling us that twice of what Josh paid is $17.38; so
what he paid must be half of that, or $8.69.
With
the equation, we can divide "both sides" by 2,
(2 J ) / 2 = 17.38 / 2
and
we get J = 8.69
From
the statement Rick paid $5 less so he paid $8.69 - $5.00 = $3.69.
But
we already wrote a formula earlier R = J - 5, as you can see,
directly
substituting 8.69 into this formula R = (8.69) - 5, we get the
same answer we reasoned to be $3.69.
So
Josh paid $8.69 and Rick paid $3.69 and a quick reread of the
question tells us we must have answered the question.
BUT
WE ARE NOT FINISHED.....
"Does
the answer make sense?"
Well
we know their contributions total $8.69 + $3.69 = $12.38, so YES,
this answer checks out and makes sense. Now, we're done.
With
these word problems, a diagram can help find their solution(s).
For
example, #4 previously:
The
length of a rectangle is 20 less than half its width. The length
is? The perimeter is?
From
here the perimeter can be visualized as a walk along the edge, once
around. We start on the bottom length, following the arrows, moving
up a width, back across a length, then down a width, so we passed 2
of each, and therefor the perimeter is the sum of 2 of each, 2L + 2W.
This is another FREE ALGEBRA PRINTABLE presented to you from the
Algebra section of
K12math.com