FORMULA REARRANGEMENT

We rearrange formulae using the same processes which we use to solve equations.

Anything we do to one side we must also do to the other.

Example 1 :

Make y the subject of :

2x + 5y = 10

Solution :

Solving the equation for y, we get

5y  =  10-2x

Divide by 5 on both sides, we get

y  =  (10-2x)/5

y  =  2 - (2x/5)

Example 2 :

Solve for x :

3x + a = d

Solution :

3x  =  d-a

x  =  (d-a)/3

Example 3 :

Solve for x :

5x + 2y = d

Solution :

5x  =  d-2y

x  =  (d-2y)/5

Example 4 :

Make a the subject of F = ma

Solution :

F = ma

a  =  F/m

Example 5 :

Make d the subject of V  =  ldh

Solution :

V  =  ldh

d  =  V/lh

Example 6 :

Make h the subject of A  =  bh/2

Solution :

A  =  bh/2

h  =  2A/b

Example 7 :

Rearrange this formula into the form y = mx + c.

Hence, state the value of :

a) the slope m

b) the y-intercept c.

Solution :

y  =  mx + c

(a) Solve the formula for m :

mx  =  y - c

m  =  (y-c)/x

(b)  Solving for c :

c  =  y-mx

Example 8 :

Make t the subject of s  =  (1/2) gt², where t > 0.

Solution :

s  =  (1/2) gt²

2s  =  gt²

t²  =  2s/g

t  =  √(2s/g)

Example 9 :

Make x the subject of y  =  (3x+2)/(x-1)

Solution :

y  =  (3x+2)/(x-1)

y(x-1)  =  3x+2

xy-y  =  3x+2

xy-3x  =  2+y

x(y-3)  =  2+y

x  =  (2+y)/(y-3)

Example 10 :

Make y the subject of M  =  4/(x²+y²)

Solution :

M  =  4/(x²+y²)

x²+y²  =  4/M

y²  =  (4/M) - x²

y  =  √[(4/M) - x²]

Example 11 :

Tilly earns p dollars for every w hours of work. Which expression represents the amount of money, in dollars, Tilly earns for 39w hours of work?

A) 39p    B) p/39    C) p + 39    D) p-39

Solution :

p dollar = w hours of work

1 hour of work = p/w

Working for 39 w hours = 39w (p/w)

= 39 p

So, the answer is option A.

Example 12 :

R = F/(N + F)

A website uses the formula above to calculate a seller's rating, R based on the number of favorable reviews. F and unfavorable reviews N, which of the following expresses the number of favorable reviews in terms of the other variables ?

a)  F = RN/(R - 1)       b)  F = RN/(1 - R)

c)  F = N/(1 - R)         d)  F = N/(R - 1)

Solution :

R = F/(N + F)

From this equation, we have to solve for the variable F. Multiply by (N + F) on both sides.

R(N + F) = F

RN + RF = F

Subtracting RF on both sides.

RN = F - RF

RN = F(1 - R)

Dividing by 1 - R on both sides, we get

F = RN/(1 - R)

Example 13 :

h = -16t² + vt +  k

The equation above gives the height h in feet, of a ball t seconds after it is thrown straight up with an initial speed of v per second from a height of k feet. Which of the following gives v in terms of h, t and k ?

a)  v = h + k - 16t           b)  v = (h - k + 16)/t

c)  v = (h + k)/t - (16t)      d)  v = (h - k)/t + 16t

Solution :

h = -16t² + vt +  k

vt = h + 16t² - k

To isolate v, we have to divide it by t on both sides.

v =  (h + 16t² - k) / t

v = [(h - k)/t] - (16t²/t)

v = [(h - k)/t] - 16t

So, option d is correct.

Example 14 :

14x/7y = 2 √(w + 19)

The given equation relates the distinct positive real numbers w, x, and y. Which equation correctly expresses w in terms of x and y ?

a)  w = √(x/y) - 19      b)  w = √(28x/14y) - 19

c)  w = (x/y)² - 19       d)  w = (28x/14y)² - 19

Solution :

14x/7y = 2 √(w + 19)

Dividing by 2 on both sides, we get

(14x/14y) = √(w + 19)

(x/y) =  √(w + 19)

Squaring on both sides, we get

(x/y)² = (w + 19)

Subtracting 19 on both sides

w = (x/y)² - 19

So, option c is correct.

Example 15 :

P = N(19 - C)

The given equation relates the positive numbers P, N, and C. Which equation correctly expresses C in terms of P and N ?

a)  C = (19 + P)/N      b)   C = (19 + P)/N

c)  C = 19 + (P/N)      d)  C = 19 - (P/N)

Solution :

P = N(19 - C)

Distributing N, we get

P = 19N - NC

Adding NC on both sides

NC = 19N - P

Dividing by N on both sides

C = 19 - (P/N)

So, option d is correct.

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