Question

If 5b = 6a + 16 and 9a = 7b - 20, then what are the values of a and b?

Answer 1

9514 1404 393

Answer:

(a, b) = (4, 8)

Explanation:

Writing both equations in general form, we have ...

6a -5b +16 = 0

9a -7b +20 = 0

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We can make a little array of the numbers in the equations:

`\left[\begin{array}{cccc}6&-5&16&6\\9&-7&20&9\end{array}\right]`

Note that the first column is repeated at the end.

Now, we can "cross multiply" and form the differences of products in adjacent columns:

d1 = (6)(-7) -(9)(-5) = 3 . . . . . . left 2 columns

d2 = (-5)(20) -(-7)(16) = 12 . . . . . middle 2 columns

d3 = (16)(9) -(20)(6) = 24 . . . . . right 2 columns

Then the solutions to these equations are the solutions to ...

1/d1 = a/d2 = b/d3

a = d2/d1 = 12/3 = 4

b = d3/d1 = 24/3 = 8

The values of a and b are 4 and 8, respectively.

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Additional Comment

There are a number of ways to solve linear equations. This is similar to what you would get by doing "elimination". It is also similar to Cramer's Rule and Vedic math techniques. The number of math operations is about the same*. Once you learn the technique, it is fairly fast and requires no decision-making. My other favorite method is using a graphing calculator.

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* This method uses 6 multiplications, 3 subtractions, 2 divisions. In the general case of "elimination", where the coefficients are not "nice" (as here), you would perform 7 multiplications, 4 subtractions and 2 divisions. This method is less work when the coefficients don't cancel easily.