To solve a single equation or inequality, isolate the variable. With several unknowns, substitute one equation into another until only one variable is left.
Isolation
Problem 8.1.1. (a)
(b)
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(a) .
(b) .
Problem 8.1.2. (a)
(b)
(c)
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(a) .
(b) .
(c) .
Problem 8.1.3. Compute and solve: .
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.
Problem 8.1.4. Compute and solve:
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Moving the first fraction across, , so
Problem 8.1.5. Find all such that
has no solution for .
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No solution is when , i.e. .
Also causes division by in the original equation. So set
no solution, so there are no values of for which . Note is also not a solution.
So and are the two values which give no solution.
Substitution and systems of equations
Problem 8.1.6.
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Substitute the third equation into the second:
Then , and the first equation becomes
With the system is
From :, so . Then , so ,, and .
Example 8.1.7.
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Idea: choose , so that only has one variable.
Or substitute: from the first equation , so
So the power of in is , i.e. and . Then
and indeed . So ,.
Problem 8.1.8. Win ratio . At the start of the weekend . Over the weekend: matches, won and lost. At the end, . What is the largest number of matches won before the weekend began?
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, and
Exercises
Exercise 8.1.9. Find all such that has no solution for .
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Exercise 8.1.10. Solve .
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Then and , giving the points and . Check: and .
Exercise 8.1.11. Solve .
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Setting this equal to and squaring:
Cancelling :
so or .
Exercise 8.1.12. A fraction , with relatively prime and positive, satisfies .
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so
Another attempt:
Exercise 8.1.13 (A count on a -gon).Show solution
On a -gon: for every vertices there exist possibilities; for every vertex there exist possible connections; .