Algebra!

I hope I didn't make any typos in the numbers…

The first problem is "One part is 2-1/4" shorter than the other" and
the second problem is "One part is 3 times as long as the other".

These equations involve x and y, so you either solve for x and then
find y, or solve for y and then find x.  I'll illustrate both.

Note - I'll write "two and one fourth" as "2-1/4" and "two times y" as
"2 * y", etc.

First problem, solving for y first then finding x
=================================================

Divide a 5-inch line into two parts: one part is 2-1/4" shorter than
the other.

Equations
---------

x + y = 5
x = y - 2-1/4

Work
----

The second equation gives us a value for x, so put it in the first
equation for x:

y - 2-1/4 + y = 5

Consolidate terms:

2 * y - 2-1/4 = 5

Add 2-1/4 to both sides:

2 * y = 5 + 2-1/4

Or, 

2 * y = 7-1/4

Divide both sides by 2 to find y:

y = 7-1/4 / 2

  = 29/4 / 2

  = 29/4 / 2/1

  = 29/4 * 1/2

  = 29/8

y = 3-5/8

So if x + y = 5, then

x = 5 - y

  = 5 - 3-5/8

x = 1-3/8

Solution:
---------

y = 3-5/8, x = 1-3/8

Check
-----

Put our values into the original equations and see what happens:

x + y = 5
x = y - 2-1/4

1-3/8 + 3-5/8 does equal 5,
1-3/8 does equal 3-5/8 - 2-1/4

First problem, solving for x first then finding y
=================================================

Divide a 5-inch line into two parts: one part is 2-1/4" shorter than
the other.

Equations
---------

x + y = 5
x = y - 2-1/4

Work
----

Add 2-1/4 to both sides of the second equation gives us an expression
for y:

y = x + 2-1/4

Put this value of y into the first equation:

x + x + 2-1/4 = 5

Subtract 2-1/4 from both sides:

x + x = 2-3/4

or,

2 * x = 2-3/4

Divide both sides by 2:

x = 2-3/4 / 2

  = 11/4 / 2

  = 11/4 * 1/2

  = 11/8

  = 1-3/8

So if x + y = 5, then

y = 5 - x

  = 5 - 1-3/8

y = 3-5/8

Solution:

x = 1-3/8, y = 3-5/8

Check
-----

Put our values into the original equations and see what happens:

x + y = 5
x = y - 2-1/4

1-3/8 + 3-5/8 does equal 5,
1-3/8 does equal 3-5/8 - 2-1/4

Second problem, solving for y first then finding x
==================================================

Divide a 5-inch line into two parts: one part is 3 times as long as
the other.

Equations
---------

x + y = 5
x = 3 * y

Work
----

We have an expression we can use for x, so use it in the first
equation:

3 * y + y = 5

Combine terms:

4 * y = 5

Divide both sides by 4 to get a number for y:

y = 5/4

y = 1-1/4

Since x = 3 * y,

x = 3 * 1-1/4

  = 3 * 5/4

  = 15/4

x = 3-3/4

Solution
--------

y = 1-1/4, x = 3-3/4

Check
-----

Put our values into the original equations and see what happens:

x + y = 5
x = 3 * y

1-1/4 + 3-3/4 does equal 5,
3 * 1-1/4 does equal 3-3/4

Second problem, solving for x first then finding y
==================================================

Divide a 5-inch line into two parts: one part is 3 times as long as
the other.

Equations
---------

x + y = 5
x = 3 * y

Work
----

Subtract x from both sides of the first equation:

y = 5 - x

Put that expression for y into the second equation:

x = 3 * (5 - x)

x = 15 - 3 * x

Add 3 * x to both sides:

4 * x = 15

Divide both sides by 4:

x = 15/4

x = 3-3/4

Since x + y = 5, 

3-3/4 + y = 5

Subtract 3-3/4 from both sides:

y = 5 - 3-3/4

y = 1-1/4

Solution
--------

x = 3-3/4, y = 1-1/4

Check
-----

Put our values into the original equations and see what happens:

x + y = 5
x = 3 * y

1-1/4 + 3-3/4 does equal 5,
3 * 1-1/4 does equal 3-3/4