9.7 Solve a formula for a specific variable  (Page 2/9)

Solve a formula for a specific variable

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 Excel ^TM 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 [link] and [link] , we used the formula $d=rt.$ This formula gives the value of $d$ when you substitute in the values of $r$ and $t.$ But in [link] , we had to find the value of $t.$ We substituted in values of $d$ and $r$ and then used algebra to solve to $t.$ If you had to do this often, you might wonder why there isn’t a formula that gives the value of $t$ when you substitute in the values of $d$ and $r.$ We can get a formula like this by solving the formula $d=rt$ for $t.$

To solve a formula for a specific variable means to get that variable by itself with a coefficient of $1$ 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.

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

1. ⓐ when $d=520$ and $r=65$
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.

	ⓐ when d = 520 and r = 65	ⓑ in general
Write the forumla.
Substitute any given values.
Divide to isolate t .
Simplify.

Notice that the solution for ⓐ is the same as that in [link] . We say the formula $t=\frac{d}{r}$ is solved for $t.$ We can use this version of the formula anytime we are given the distance and rate and need to find the time.

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

1. ⓐ when $A=90$ and $b=15$
2. ⓑ in general

Solution

	ⓐ when A = 90 and b = 15	ⓑ in general
Write the forumla.
Substitute any given values.
Clear the fractions.
Simplify.
Solve for h .

We can now find the height of a triangle, if we know the area and the base, by using the formula

$h=\frac{2A}{b}$

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