1130

Solve for x the equation 3x + 7 = 13.

Starting with

3x + 7 = 13 we first add -7 to both sides:

3x + 7 + (-7) = 13 + (-7). 

Since -7 is the additive inverse of 7 the left-hand side becomes 3x + 0, and since 0 is the additive identity the left-hand side is just 3x.

The right-hand side is just 13 - 7 = 6.  So our equation becomes

3x = 6.

Since 1/3 is the multiplicative inverse of 3 we multiply both sides by 1/3 to get

1/3 ( 3 x) = 1/3 ( 6 ).

The left-hand side is therefore 1/3 ( 3 x ) = 1/3 * 3 * x = 1 * x.  Since 1 is the multiplicative identity the left-hand side becomes x.  The right-hand side becomes 6 / 3 = 2.  So our equation becomes

x = 2.

Solve for x: 

3 / 2 ( 7x - 5) - 1 = 5 /3 ( 4x + 3).

First we make sure we understand what all the terms in this expression mean.

3 / 2 ( 7x - 5) means 3 / 2 * (7 x - 5).  This expression indicates a division followed by a multiplication.  The division and the multiplication must be done in order, so this expression really means (3/2) * (7x - 5).  The (7x - 5) is not part of an implied denominator, and is never multiplied by the 2.

Similarly 5 /3 ( 4x + 3) means (5/3) ( 4 x + 3).

It is important to understand that when you type a mathematical expression into a computer algebra system, the expression will be interpreted strictly according to the order of operations.

Now, expressing the equation as

(3/2) ( 7 x - 5) - 1 = (5/3) ( 4 x + 3)

The equation has two denominators, 2 and 3.  The LCM of these denominators is 6 so we multiply both sides of the equation by 6 to get

6 * (3/2) ( 7 x - 5) - 6 * 1 = 6 * (5/3) ( 4 x + 3).

This can be rearranged to give us

(6/2) * 3 ( 7x - 5) - 6 = (6/3) * 5 ( 4x + 3)

or

3 * 3 ( 7x - 5) - 6 = 2 * 5 ( 4x + 3) or

9 ( 7x - 5) - 6 = 10 ( 4x + 3).

Now that fractions have been cleared we use the distributive law to write

63 x - 45 - 6 = 40 x + 30.  Simplifying the left-hand side we have

63 x - 51 = 40 x + 30.  Adding 51 to both sides and applying inverse and identity properties for addition (see last example for details and be sure you understand these details) we end up with

63 x = 40 x + 81.  Adding -40 x to both sides and applying inverse and identity properties for addition we get

23 x = 81.  Finally multiplying both sides by 1/23 and applying inverse and identity properties for multiplication we get

x = 81/ 23.

Solve the formula A = 2 pi r h for r.

We can multiply both sides by 1/2:

1/2 ( A ) = 1/2 ( 2 pi r h ).  This would give us

A / 2 = 1/2 * 2 ( pi r h), or

A / 2 = (2/2) ( pi r h) and since 2 is the multiplicative inverse of 2 we have

A / 2 = 1 ( pi r h).  Since 1 is the multiplicative identity we have

A / 2 = pi r h.

This is progress.  We now have less stuff hangin' on the r than we did before.

We could then divide by pi, or by h, to get closer to isolating r.  In fact, we can divide by (pi * h) all at once.  To justify this, we multiply both sides by 1 / ( pi * h) to get

1 / ( pi * h) * A / 2 = 1 / ( pi * h) * pi r h.

This can be expressed at

A / ( 2 pi h) = ( pi h) / (pi h) * r,

and since pi h / ( pi h) = 1, the multiplicative identity, we have

A / ( 2 pi h ) = r.

Reversing sides we have

r = A / ( 2 pi h).

Note that we could have mutliplied the original equation by 1 / ( 2 pi h) to get this result more quickly.

Solve the equation

A = ( h1 + h2 ) / 2 * w

for h2.

Multiplying both sides by 2 and justifying everything in terms of inverses, identities etc. we get

2 * A = (h1 + h2) * w.

Multiplying both sides by 1/w and justifying as usual we get

2 * A / w = h1 + h2.

Adding -h1 to both sides and justifying everything we have

2 * A / w - h1 = h2 so

h2 = 2 * A / 2 - h1.