TRANSFORMING FORMULAS

Formulas are being used in many applications of mathematics.

For example, we use the formula A  =  πr² to find area of a circle. It is often helpful to transform such a formula to express a particular variable in terms of the other variables. 

Example 1 : 

A formula for the total piston displacement of an automobile engine is 

P  =  0.7854d²sn

where d is the diameter of each cylinder, s is the length of the stroke, and n is the number of cylinders. Solve for the variable s in terms of P, d and n. 

Solution : 

Write the given formula. 

P  =  0.7854d²sn

Because we are solving for s, we have to get s alone on one side. 

So, divide each side by 0.7854d²n.

P / 0.7854d²n  =  s

Note : 

Note that the formula obtained for s in the above example is valid only if d ≠ 0 and n ≠ 0. (Of course, neither n nor d will be zero, since the engine must have cylinders and each cylinder must have a diameter).    

Example 2 : 

Solve the formula for the variable r :  

A  =  s² + 2rs

Solution : 

Write the given formula. 

A  =  s² + 2rs 

Because we are solving for r, we have to get r alone on one side. 

Subtract s² from each side.  

A - s²  =  2rs

Divide each side by 2s.

(A - s²) / 2s  =  r

Note : 

Note that the formula obtained for s in the above example is valid only if s ≠ 0. And also, if s is the length of a side, then s > 0.    

Example 3 : 

Solve the equation for the variable x :  

c  =  ax - b

Solution : 

Write the given equation. 

c  =  ax - b 

Because we are solving for x, we have to get x alone on one side. 

Add b to each side.   

b + c  =  ax

Divide each side by a.

(b + c) / a  =  x

Example 4 : 

Solve the equation for the variable x :  

C  =  mv² / r

Solution : 

Write the given equation. 

C  =  mv² / r

Because we are solving for r, we have to get r alone on one side. 

Take reciprocal on each side.

1 / C  =  r / mv²

Multiply each side by mv².

mv² ⋅ (1 / C)  =  (r / mv²) ⋅ mv²

mv² / C  =  r

Example 5 : 

Solve the given formula for the variable h. State the restrictions, if any, for the formula obtained to be meaningful.      

A  =  1/2 ⋅ h ⋅ (a + b)

Solution : 

Write the given formula. 

A  =  1/2 ⋅ h ⋅ (a + b)

Because we are solving for h, we have to get h alone on one side. 

Multiply each side by 2. 

2A  =  [1/2 ⋅ h ⋅ (a + b)] ⋅ 2

2A  =  h ⋅ (a + b)

Divide each side by (a + b).

2A / (a + b)  =  h

Note that the formula obtained for h in the above example is valid only if (a + b)  ≠ 0. (If a and b are the lengths of the sides, then a, b > 0 and (a + b) > 0).    

Example 6 : 

Solve the given formula for the variable F. State the restrictions, if any, for the formula obtained to be meaningful.      

C  =  5/9 ⋅ (F - 32)

Solution : 

Write the given formula. 

C  =  5/9 ⋅ (F - 32)

Because we are solving for F, we have to get F alone on one side. 

Multiply each side by 9/5. 

9/5 ⋅ C  =  [5/9 ⋅ (F - 32)] ⋅ 9/5

9C/5  =  F - 32

Add 32 to each side. 

9C/5 + 32  =  F

Example 7 : 

Solve the given formula for the variable y. State the restrictions, if any, for the formula obtained to be meaningful.      

m  =  (x + y + z) / 3

Solution : 

Write the given formula. 

m  =  (x + y + z) / 3

Because we are solving for y, we have to get y alone on one side. 

Multiply each side by 3. 

3m  =  [(x+ y + z) / 3] ⋅ 3

3m  =  x + y + z

Subtract x and z from each side.

3m - x - z  =  y

Example 8 : 

Solve the given formula for the variable n. State the restrictions, if any, for the formula obtained to be meaningful.      

a  =  180(n - 2) / n

Solution : 

Write the given formula. 

a  =  180(n - 2) / n

Because we are solving for n, we have to get n alone on one side. 

Multiply each side by n. 

na  =  [180(n - 2) / n] ⋅ n

na  =  180(n - 2)

na  =  180n - 360

Add 360 to each side. 

na + 360  =  180n

Subtract na from each side. 

360  =  180n - na

360  =  (180 - a)n

Divide each side by (180 - a).

360 / (180 - a)  =  n

Note that the formula obtained for n in the above example is valid only if (180 - a)  ≠ 0, that is  a  ≠  180. And also, if n is the number of sides, then n > 0. Then we have (180 - a) > 0, that is a < 180.    

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