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Answer:
x = 30
y = 68
z = 82
Step-by-step explanation:
Let the first angle be X, the second angle be Y, and the third angle be Z.
You can write these 3 equations to represent the relationships between the angles:
[tex]x+y+z=180\\y+z=5x\\z=y+14[/tex]
Step 1. Z is already isolated in the 3rd equation, so the first thing I would do is use that to eliminate the Z in the 1st and 2nd equations.
[tex]z=y+14\\\\x+y+z=180\\x+y+(y+14)=180\\\\y+z=5x\\y+(y+14)=5x[/tex]
Step 2. Now, simplify both of those and treat them like a system of 2 equations. I'll simplify the top equation first and solve for x:
[tex]x+y+(y+14)=180\\x+y+y+14=180\\x+2y+14=180\\x+2y=166\\x=166-2y[/tex]
Plug that into the other equation:
[tex]y+(y+14)=5x\\y+y+14=5(166-2y)\\2y+14=830-10y\\12y+14=830\\12y=816\\y=68[/tex]
Step 3. Now we have the value of one variable. Use that to solve for x in the previous equation:
[tex]x=166-2y\\x=166-2(68)\\x=166-136\\x=30[/tex]
Step 4. Now we have two variables. Pick any of the original 3 equations you want to solve for z:
[tex]x+y+z=180\\30+68+z=180\\98+z=180\\z=82[/tex]
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Finally, you can confirm that by plugging these into all of the original equations to make sure they meet the conditions.
[tex]30+68+82=180\\180=180\\\\68+82=5(30)\\150=150\\\\82=68+14\\82=82[/tex]