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The following are some typical problems that, in a natural way, lead us to algebraic equations. In each problem, the solution to the problem is found
by finding the solution set of an equation. When we find the solution set of an equation, we say that we have solved the equation.
Example 1: A number has such a property that when 6 is subtracted from twice the number, the result is 16. What is the number?
Solution: The required number must satisfy the open sentence
or, equivalently, the required number must satisfy each of the following open sentences:
Since the only number that satisfies the last equation is 11, 11 is the required number. We may check our result:
Example 2: A number has such a property that when 10 is subtracted from twice the number, the result is the same as when 6 is added to the number. What is the number?
Solution: A number has the required property if and only if it satisfies the open sentence
2 - x - 10 = x + 6
or, equivalently, a number has the required property if and only if it satisfies each of the following open sentences:
(2 - x - 10) + 10 = (x + 6) + 10
2x = x + 16
- x + 2x = - x + (x + 16)
(- x + x) + x = (- x + x)+ 16
x = 16
The only number that satisfies the last open sentence is 16. Therefore, 16 is the required number.
Example 3: Suppose that Tom has $6.00 more than Bill and that together they have a total of $12.00. How much money does Tom have? How much money does Bill have?
Solution: Suppose that Tom has 't' dollars and that Bill has 'b' dollars. Since Tom has $6.00 more than Bill, then
(1) t = 6 + b
Since Tom and Bill have together a total of $12.00, then
(2) t + b = 12
Equation (1) tells us that
t is the same as 6 + b
We may therefore replace 't' by '6 + b' in equation (2):
(6 + b) + b = 12
6 + 2b = 12
2b = 6
b = 3
Therefore Bill has $3.00. If follows that
t = 6 + b = 6 + 3 = 9
and so Tom has $9.00.
(1) 9 = 6 + 3
(2) 9 + 3 = 12
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