Answer:
y = {-180, 2 8/11}
Step-by-step explanation:
I like to solve these using a graphing calculator. I generally start by writing the equation in the form ...
f(y) = 0
This can be done by subtracting the right-side expression from both sides. The solution will be the horizontal intercepts of the graph:
y = -180; y = 2 8/11
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To solve this analytically, you need to recognize that this resolves into four equations, effectively two identical pairs. Each equation comes with a set of conditions (domain) for which it is applicable. Here is the expanded set of equations:
1/3y -7 = 2/5y +5 . . . both positive: 1/3y -7 ≥ 0 and 2/5y + 5 ≥ 0
7 -1/3y = 2/5y +5 . . . left side negative: 1/3y -7 ≤ 0 and 2/5y +5 ≥ 0
1/3y -7 = -2/5y -5 . . . right side negative: 1/3y -7 ≥ 0 and 2/5y +5 ≤ 0
7 -1/3y = -2/5y -5 . . . both sides negative: 1/3y -7 ≤ 0 and 2/5y +5 ≤ 0
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The conditions resolve to ...
1/3y -7 ≥ 0
y -21 ≥ 0 . . . . . multiply by 3
y ≥ 21 . . . . . . . add 21
and ...
2/5y +5 ≥ 0
y +25/2 ≥ 0 . . . . multiply by 5/2
y ≥ -25/2 . . . . . subtract 25/2
Now, we are in a position to solve the equations and determine where the solution is applicable.
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both positive
The domain will be y ≥ 21 and y ≥ -25/2, which is y ≥ 21.
Multiplying the equation by 15 gives ...
5y -105 = 6y +75
y = -180 . . . . . . . . . subtract 5y+75
This solution is not in the allowed domain.
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left side negative
The domain will be y ≤ 21 and y ≥ -25/2, which is -25/2 ≤ y ≤ 21.
Multiplying the equation by 15 gives ...
105 -5y = 6y +75
30 = 11y . . . . . . . . . add y-75
y = 30/11 . . . . . . the solution is in the allowed domain
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right side negative
The domain will be y ≥ 21 and y ≤ -25/2. This is the empty set.
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both sides negative
The domain will be y ≤ 21 and y ≤ -25/2, which is y ≤ -25/2.
Multiplying the equation by 15 gives ...
105 -5y = -6y -75
y = -180 . . . . . . . . . . add 6y-105 . . . . the solution is in the allowed domain
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Solutions to this equation are ...
y = 30/11 = 2 8/11
y = -180
