Answer: x = 7 ; y = 4 .
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sin 30 / (5y - 4) = sin 30 / (y + 12) ;
(sin 60) / (5y - 4) = (sin 60) / (3x - 5) ;
5y - 4 = 3x - 5 ;
5y - 4 = 3x - 5 = y + 12 ;
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5y - 4 = y + 12 ; solve for "y" ;
Add "4" to each side of the equation; & Subtract "5y" from each side of the equation:
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5y - 4 + 4 - 5y = y + 12 + 4 - 5y ;
to get: 0 = -4y + 16 ;
↔ -4y + 16 = 0 ;
Subtract "16" from each side of the equation:
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-4y + 16 - 16 = 0 - 16 ;
to get: -4y = -16 ;
Now, divide EACH SIDE of the equation by "-4" ; to isolate "y" on each side of the equation; and to solve for "y" :
-4y / -4 = -16 / -4 ;
to get: " y = 4 " ,
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Now, to solve for "x" :
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Since:
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5y - 4 = 3x - 5 ;
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Substitute "4" for "y" (in the equation); and solve for "x" ;
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(5*4) - 4 = 3x - 5 ;
20 - 4 = 3x - 5 ;
16 = 3x - 5 ;
↔ 3x - 5 = 16 ;
Add "5" to each side of the equation;
3x - 5 + 5 = 16 + 5 ;
3x = 21 ;
Now divide EACH SIDE of the equation by "3" ; to isolate "x" on ONE SIDE of the equation; and to solve for "x" ;
3x / 3 = 21 / 3 ;
x = 7 ;
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Let us check our work:
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Given the equation:
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" 5y - 4 = 3x - 5 " ;
→ Does the equation hold true when "x = 7" and "y = 4" ?
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→ On the "left-hand side" of the equation:
"5y - 4" = 5(4) - 4 = 20 - 4 = 16 .
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On the "right-hand side" of the equation:
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3x - 5 = 3(7) - 5 = 21 - 5 = 16 .
Does "16 = 16" ? Yes! .
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Also, note: "5y - 4 = 3x - 5 = y + 12 ".
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Does: "y + 12 = 16" ; when "y = 4" ?
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→ y + 12 = 4 + 12 = 16. Yes!
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Our values: x = 7 , y = 4 .
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