Solving a Formula for a Specific Variable

Learning Outcomes

  • Solve a formula or equation for a specific variable using the properties of equality


In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft ExcelTM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.

In some examples, we used the formula
d=rtd=rt
. This formula gives the value of
dd
when you substitute in the values of
rr
and
tt
. But in another example, we had to find the value of
tt
. We substituted in values of
dd
and
rr
and then used algebra to solve for
tt
. If you had to do this often, you might wonder why there isn’t a formula that gives the value of
tt
when you substitute in the values of
dd
and
rr
. We can get a formula like this by solving the formula
d=rtd=rt
for
tt
.

To solve a formula for a specific variable means to get that variable by itself with a coefficient of
11
on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.
Let’s try a few examples, starting with the distance, rate, and time formula we used above.

example

Solve the formula
d=rtd=rt
for
t:t\text{:}


  1. When
    d=520d=520
    and
    r=65r=65
  2. In general.


Solution:

We’ll write the solutions side-by-side so you can see that solving a formula in general uses the same steps as when we have numbers to substitute.

1. When d = 520 and r = 65 2. In general
Write the formula.
d=rtd=rt
d=rtd=rt
Substitute any given values.
520=65t520=65t
Divide to isolate t.
52065=65t65\frac{520}{65}=\frac{65t}{65}
dr=rtr\frac{d}{r}=\frac{rt}{r}
Simplify.
8=t8=t


t=8t=8
dr=t\frac{d}{r}=t


t=drt=\frac{d}{r}
We say the formula
t=drt=\frac{d}{r}
is solved for
tt
. We can use this version of the formula any time we are given the distance and rate and need to find the time.

We used the formula
A=12bhA=\frac{1}{2}bh
to find the area of a triangle when we were given the base and height. In the next example, we will solve this formula for the height.

example

The formula for area of a triangle is
A=12bhA=\frac{1}{2}bh
. Solve this formula for
h:h\text{:}


  1. When
    A=90A=90
    and
    b=15b=15
  2. In general




 

Previously, we used the formula
I=PrtI=Prt
to calculate simple interest, where
II
is interest,
PP
is principal,
rr
is rate as a decimal, and
tt
is time in years.

example

Solve the formula
I=PrtI=Prt
to find the principal,
P:P\text{:}


  1. When
    I=$5,600,r=4%,t=7yearsI=\text{\$5,600},r=\text{4\%},t=7\text{years}
  2. In general




 

Watch the following video to see another example of how to solve an equation for a specific variable.



Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually
xx
and
yy
. You might be given an equation that is solved for
yy
and you need to solve it for
xx
, or vice versa. In the following example, we’re given an equation with both
xx
and
yy
on the same side and we’ll solve it for
yy
. To do this, we will follow the same steps that we used to solve a formula for a specific variable.

example

Solve the formula
3x+2y=183x+2y=18
for
y:y\text{:}


  1. When
    x=4x=4
  2. In general




 

In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide?

example

Solve the formula
P=a+b+cP=a+b+c
for
aa
.



 

example

Solve the equation
3x+y=103x+y=10
for
yy
.



 

example

Solve the equation
6x+5y=136x+5y=13
for
yy
.



 

In the following video we show another example of how to solve an equation for a specific variable.



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