Sketch of Triangle with unknowns h and y assigned

Use:       Pythagorean theorem: C² = A² + B²

First right triangle:

h² + y² = 13²

h² + y² - y² = 13² - y² Subtract y² from both sides

Gives:            h² = 13² - y²
 
 

Second right triangle:

h² + (15 - y) ² = 14²

h² + (15 - y) ² - (15 - y) ² = 14² - (15 - y)²                     Subtract (15 - y) ² from both sides

Gives:           h² = 14² - (15 - y) ²

Now substitute h² from first equation gives:

13² - y² = 14² - (15 - y) ²       Expand (15 - y) ²

          15 - y
Times 15 - y
        - 15y + y²                     Multiply 15 - y by - y               Notice -y times -y is +y²
225  - 15y ____                   Multiply 15 - y by 15
225 - 30y + y²

13² - y² = 14² – (225 - 30y + y²)                      Remove Parens

13² - y² = 14² – 225 + 30y - y²                           Add y² to both sides i.e. delete - y² from both sides

13² = 14² – 225 + 30y                                           Multiply squared terms i.e. 13² and 14²

169 = 196 – 225 + 30y                                           Move constants to same side of equation

169 + 225 - 196 = +30y                                         Add constants

198 = 30y                                                               Divide by 30

y = 6.6