Let aA+bB+C=0, where A=(29, -1.6), B=(-11, 38), and C= (82, 92). What is the value of a, and value of b?

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txmedteach | High School Teacher | (Level 3) Associate Educator

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Just to clarify, I'm going to rewrite the question with the math editor to reflect the vector nature of the problem:

`a[[29],[-1.6]] + b[[-11],[38]] + [[82],[92]] = [[0],[0]]`

A way to rewrite this would be to use matrices and to subtract C from both sides:

`[[29,-11],[-1.6, 38]][[a],[b]] = [[-82],[-92]]`

Now, we are set up to use Cramer's rule, where given a matrix equation Mx = V, with the following values for each component:

`M = [[29, -11],[-1.6, 38]]`` `

`x = [[a],[b]]`

`V = [[-82],[-92]]`

We can find the ith element of the vector, x, by doing the following (See link):

`x_(i) = det(M_(i))/det(M)`

Where `M_(i)` is the matrix M with the ith column replaced by the vector V.

In our example, `x_(1)` = a, and `x_(2)` = b. So, we get the following answers for a and b:

`a = det(M_(1))/det(M)`

`b = det(M_(2))/det(M)`

We need to get `M_(1)` and `M_(2)` by replacing the first and second columns, respectively, with our V vector from above:

`M = [[29, -11],[-1.6, 38]] V = [[-82],[-92]]`

`M_(1) = [[-82, -11],[-92, 38]]`

`M_(2) = [[29, -82], [-1.6, -92]]`

Now, we can get our determinants:

`det(M) = 29(38) - (-11)(-1.6) = 1102 - 17.6 = 1084.4`

`det(M_(1)) = -82(38) - (-11)(-92) = -3116 - 1012 = -4128`

`det(M_(2)) = 29(-92)-(-82)(-1.6) = -2668 - 131.2 = -2799.2`

Using these determinants, we can find the values for a and b:

`a = det(M_(1))/det(M) = -4128/1084.4 = -3.81`

`b = det(M_(2))/det(M) = -2799.2/1084.4 = -2.58`

There are plenty of other ways to do this. For example, we could have found the inverse of `M` (`M^-1`) and done the following multiplication:

`M^-1Mx = M^-1V`

`x = M^-1V`

Which may be easier to you, considering the inverse is easy to calculate as seen below (see the linked invertible matrix article):

`M^-1 = 1/det(M)[[38, 11],[1.6, 29]]`

Finally, a way to think about this that goes back to Algebra I is to notice that with those vectors, you're creating a system of 2 equations:

`29a - 11b + 82 = 0`

`-1.6a + 38b + 92 = 0`

You could then solve by substitution or elimination algebraically, or you could graph the lines by letting x = a and y = b so you can find the intersection.

So there, 5 ways (bolded methods in answer) to get the final result:

a = -3.81, b = -2.58

Sources:

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