370 second song is divided into three parts, the beginning, the middle, and the end. the beginning is one sixth as long as the end. the middle is one half as long as the end. find the length of each part

Step-by-step explanation:

The main part of the song is the end.  The beginning length is based on the end and so is the middle length.  The beginning length is 1/6 as long as the end, so if the end is our main unknown, x, the beginning is 1/6x.  The middle is one-half as long as the end, so if the end is our main unknown, x, the middle is 1/2x.  The sum of the 3 parts is 370, so:

The GCF of those denominators is 6, so multiply through every term by 6 to get rid of the fractions:

Simplifying gives us

6x + x + 3x = 2220 and

10x = 2220 so

x = 222

That means that the end is 222 seconds, the bginning is 37 seconds, and the middle is 111 seconds.

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.