Answer:
Step-by-step explanation:
Let "a" and "b" represent the first and second question marks, respectively. Then you want to find "a" and "b" such that this equation is true for all values of p:
(2/5)p + (5/6) = (a·p +1/2) - ((3/5)p -b)
If we simplify the right side of this equation, we get ...
(2/5)p + (5/6) = (a -3/5)·p + (1/2 +b)
In order for this to be true for all values of p, <em>the coefficients of p must match, and the constant terms must match</em>. That is, ...
- 2/5 = a - 3/5 . . . . . . coefficients of p
- 5/6 = 1/2 + b . . . . . . constant terms
These two equations are easily solved in the usual way.
2/5 + 3/5 = a - 3/5 + 3/5 . . . . . add 3/5 to both sides of the first equation
5/5 = a = 1 . . . . . simplify
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5/6 - 1/2 = 1/2 - 1/2 + b . . . . . . subtract 1/2 from both sides of the second equation
5/6 - 3/6 = b = 2/6 = 1/3 . . . . . simplify
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The first question mark is filled with 1; the second question mark is filled with 1/3.
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<em>Comment on this problem</em>
There are many kinds of problems that are solved by "matching coefficients." That is, you identify the coefficient of a particular variable on one side of the equal sign, and you make the coefficient of that variable on the other side of the equal sign match it (have the same value).
Here, we're matching the coefficients of "p", which are 2/5 and (a-3/5). Likewise, we're matching the constants, 5/6 and (1/2 -(-b)). The problem presumes you are comfortable doing arithmetic with fractions and minus signs.
