y = $\sqrt{\frac{\mathrm{px}}{\mathrm{r}}-{\mathrm{r}}^2\mathrm{x}}$

Get rid of the square root by squaring both sides of the equation.

y² = $\frac{\mathrm{px}}{\mathrm{r}}-{\mathrm{r}}^2\mathrm{x}$

Factorize x out.

y² = x($\frac{\mathrm{p}}{\mathrm{r}}-{\mathrm{r}}^2$)

Every number is being divided by 1

r² = $\frac{{\mathrm{r}}^2}{1}$

y² = x($\frac{\mathrm{p}}{\mathrm{r}}-\frac{{\mathrm{r}}^2}{1}$)

y² = x($\frac{\mathrm{p\; -}\mathrm{r\; x}{\mathrm{r}}^2}{\mathrm{r}}$)

Note: r x r² = r^{1 + 2} = r³

y² = x($\frac{\mathrm{p\; -}{\mathrm{r}}^3}{\mathrm{r}}$)

Multiply both sides by r to get rid of the fraction.

y² x r = x(p - r³)

ry² = x(p - r³)

Divide both sides by (p - r³)

x = $\frac{\mathrm{r}{\mathrm{y}}^2}{\mathrm{p}-{\mathrm{r}}^3}$

$\frac{\mathrm{t}}{\mathrm{s\; +\; u}}$ = $\frac{\mathrm{s}}{\mathrm{t\; -\; u}}$

Cross multiply

t x (t - u) = s x (s + u)

t² - tu = s² + su

t² - s² = tu + su

t² - s² = u(t + s)

Divide both sides by (t + s)

u = $\frac{{\mathrm{t}}^2-{\mathrm{s}}^2}{\mathrm{t\; +\; s}}$

From difference of two squares, a² - b² = (a + b)(a - b)

u = $\frac{\mathrm{(t\; +\; s)(t\; -\; s)}}{\mathrm{t\; +\; s}}$

t + s cancels out

u = t - s

A = $\frac{1}{2}$b(a + b)

Get rid of the fraction by multiplying both sides by 2

2 x A = 1 x b(a + b)
2A = b(a + b)

Divide both sides by b

$\frac{\mathrm{2A}}{\mathrm{b}}$ = a + b

$\frac{\mathrm{2A}}{\mathrm{b}}$ - b = a

x = $\frac{\mathrm{2U\; -\; 3}}{\mathrm{3U\; +\; 2}}$

Get rid of the fraction by multiplying each sides by the denominator (3U + 2).

x x (3U + 2) = 2U - 3

3xU + 2x = 2U - 3

3xU - 2U = -2x - 3

Factorize U out.

U(3x - 2) = -2x - 3

Divide both sides by (3x - 2).

U = $\frac{\mathrm{-2x\; -\; 3}}{\mathrm{3x\; -\; 2}}$

Factorize (-1) out.

U = $\frac{-1}{-1}$ x $\frac{\mathrm{2x\; +\; 3}}{\mathrm{-3x\; +\; 2}}$

-1 cancels -1 and you can rearrange -3x + 2 as 2 - 3x

U = $\frac{\mathrm{2x\; +\; 3}}{\mathrm{2\; -\; 3x}}$

Alternatively,

3xU + 2x = 2U - 3

2x + 3 = 2U - 3xU

Factorize U out.

2x + 3 = U(2 - 3x)

Divide both sides by (2 - 3x)

$\frac{\mathrm{2x\; +\; 3}}{\mathrm{2\; -\; 3x}}$ = U

U = $\frac{\mathrm{2x\; +\; 3}}{\mathrm{2\; -\; 3x}}$

k = $\sqrt{\frac{\mathrm{m\; -\; y}}{\mathrm{m\; +\; 1}}}$

Square both sides to get rid of the square root.

k² = $\frac{\mathrm{m\; -\; y}}{\mathrm{m\; +\; 1}}$

Get rid of the fraction by multiplying both sides by the denominator m + 1

k² x (m + 1) = m - y

mk² + k² = m - y

y + k² = m - mk²

Factorize m out.

y + k² = m(1 - k²)

Divide both sides by (1 - k²)

m = $\frac{\mathrm{y}+{\mathrm{k}}^2}{1-{\mathrm{k}}^2}$

lb = $\frac{1}{2}$(a + b)h

Get rid of the fraction by multiplying both sides by the denominator 2

lb x 2 = h(a + b)

2lb = ah + bh

2lb - bh = ah

Factorize b out.

b(2l - h) = ah

Divide both sides by (2l - h)

b = $\frac{\mathrm{ah}}{\mathrm{2l\; -\; h}}$

E = $\frac{\mathrm{k}{\mathrm{x}}^2}{\mathrm{2y}}$ + z

Multiply both sides the denominator 2y

E x 2y = $\frac{\mathrm{k}{\mathrm{x}}^2}{\mathrm{2y}}$ x 2y + z x 2y

2Ey = kx² + 2yz

kx² = 2Ey - 2yz

Divide both sides by k.

x² = $\frac{\mathrm{2Ey\; -\; 2yz}}{\mathrm{k}}$

Factorize 2y out.

x² = $\frac{\mathrm{2y(E\; -\; z)}}{\mathrm{k}}$

Take square root of both sides.

x = $\sqrt{\frac{\mathrm{2y(E\; -\; z)}}{\mathrm{k}}}$

3^-x = k

a^-b = $\frac{1}{{\mathrm{a}}^{\mathrm{b}}}$

3^-x = $\frac{1}{3^{\mathrm{x}}}$

$\frac{1}{3^{\mathrm{x}}}$ = k

Make 3^x the subject.

Get rid of the fraction by multiplying both sides by the denominator, 3^x

3^x x $\frac{1}{3^{\mathrm{x}}}$ = k x 3^x

1 = k x 3^x

Divide both sides by k.

$\frac{1}{\mathrm{k}}$ = 3^x

3^x = $\frac{1}{\mathrm{k}}$

$\frac{\mathrm{t}}{\mathrm{s\; +\; u}}$ = $\frac{\mathrm{s}}{\mathrm{t\; -\; u}}$

Cross multiply

t x (t - u) = s x (s + u)

t² - tu = s² + su

t² - s² = tu + su

t² - s² = u(t + s)

Divide both sides by (t + s)

u = $\frac{{\mathrm{t}}^2-{\mathrm{s}}^2}{\mathrm{t\; +\; s}}$

From difference of two squares, a² - b² = (a + b)(a - b)

u = $\frac{\mathrm{(t\; +\; s)(t\; -\; s)}}{\mathrm{t\; +\; s}}$

t + s cancels out

u = t - s

P ∝ $\frac{{\mathrm{v}}^2}{\mathrm{x}}$

Replace x by vt

P ∝ $\frac{{\mathrm{v}}^2}{\mathrm{vt}}$

v² = v x v

P ∝ $\frac{\mathrm{v\; x\; v}}{\mathrm{vt}}$

The v cancel each other.

P ∝ $\frac{\mathrm{v}}{\mathrm{t}}$

A = $\frac{1}{2}$b(a + b)

Get rid of the fraction by multiplying both sides by 2

2 x A = 1 x b(a + b)
2A = b(a + b)

Divide both sides by b

$\frac{\mathrm{2A}}{\mathrm{b}}$ = a + b

$\frac{\mathrm{2A}}{\mathrm{b}}$ - b = a

x = $\frac{\mathrm{2U\; -\; 3}}{\mathrm{3U\; +\; 2}}$

Get rid of the fraction by multiplying each sides by the denominator (3U + 2).

x x (3U + 2) = 2U - 3

3xU + 2x = 2U - 3

3xU - 2U = -2x - 3

Factorize U out.

U(3x - 2) = -2x - 3

Divide both sides by (3x - 2).

U = $\frac{\mathrm{-2x\; -\; 3}}{\mathrm{3x\; -\; 2}}$

Factorize (-1) out.

U = $\frac{-1}{-1}$ x $\frac{\mathrm{2x\; +\; 3}}{\mathrm{-3x\; +\; 2}}$

-1 cancels -1 and you can rearrange -3x + 2 as 2 - 3x

U = $\frac{\mathrm{2x\; +\; 3}}{\mathrm{2\; -\; 3x}}$

Alternatively,

3xU + 2x = 2U - 3

2x + 3 = 2U - 3xU

Factorize U out.

2x + 3 = U(2 - 3x)

Divide both sides by (2 - 3x)

$\frac{\mathrm{2x\; +\; 3}}{\mathrm{2\; -\; 3x}}$ = U

U = $\frac{\mathrm{2x\; +\; 3}}{\mathrm{2\; -\; 3x}}$

k = $\sqrt{\frac{\mathrm{m\; -\; y}}{\mathrm{m\; +\; 1}}}$

Square both sides to get rid of the square root.

k² = $\frac{\mathrm{m\; -\; y}}{\mathrm{m\; +\; 1}}$

Get rid of the fraction by multiplying both sides by the denominator m + 1

k² x (m + 1) = m - y

mk² + k² = m - y

y + k² = m - mk²

Factorize m out.

y + k² = m(1 - k²)

Divide both sides by (1 - k²)

m = $\frac{\mathrm{y}+{\mathrm{k}}^2}{1-{\mathrm{k}}^2}$

lb = $\frac{1}{2}$(a + b)h

Get rid of the fraction by multiplying both sides by the denominator 2

lb x 2 = h(a + b)

2lb = ah + bh

2lb - bh = ah

Factorize b out.

b(2l - h) = ah

Divide both sides by (2l - h)

b = $\frac{\mathrm{ah}}{\mathrm{2l\; -\; h}}$

x = $\frac{\mathrm{mn}}{3}$ ------ eqn 1

Substitute m = $\frac{\mathrm{v}}{\mathrm{y}}$ into ------ eqn 1

x = $\frac{\mathrm{v}}{\mathrm{y}}$ x n ÷ 3

x = $\frac{\mathrm{v\; x\; n}}{\mathrm{y}}$ ÷ 3

x = $\frac{\mathrm{vn}}{\mathrm{y}}$ ÷ 3

Every number is being divided by 1 but it is not written.

3 = $\frac{3}{1}$

x = $\frac{\mathrm{vn}}{\mathrm{y}}$ ÷ $\frac{3}{1}$

Reciprocate the fraction at the right of the ÷ and change ÷ to multiplication (x)

Note: reciprocate means the numerator becomes the denominator and the denominator becomes the numerator.

x = $\frac{\mathrm{vn}}{\mathrm{y}}$ x $\frac{1}{3}$

x = $\frac{\mathrm{vn\; x\; 1}}{\mathrm{y\; x\; 3}}$

x = $\frac{\mathrm{vn}}{\mathrm{3y}}$

E = $\frac{\mathrm{k}{\mathrm{x}}^2}{\mathrm{2y}}$ + z

Multiply both sides the denominator 2y

E x 2y = $\frac{\mathrm{k}{\mathrm{x}}^2}{\mathrm{2y}}$ x 2y + z x 2y

2Ey = kx² + 2yz

kx² = 2Ey - 2yz

Divide both sides by k.

x² = $\frac{\mathrm{2Ey\; -\; 2yz}}{\mathrm{k}}$

Factorize 2y out.

x² = $\frac{\mathrm{2y(E\; -\; z)}}{\mathrm{k}}$

Take square root of both sides.

x = $\sqrt{\frac{\mathrm{2y(E\; -\; z)}}{\mathrm{k}}}$