Among all pairs of numbers (x, y) such that 3x+y=15, find the pair for which the sum of squares, x^2+y^2, is minimum. Write your answers as fractions reduced to lowest terms.

Step-by-step explanation:

Lets substitute y in the second equation and find its minimum

Substitute

This expression gets minimum value at x = -b/2a as quadratic function's vertex:

Then finding the value of y:

The answer is (4.5, 1.5)

Step-by-step explanation:

We have the equation:

Which is equivalent to 15 among all pairs of (x, y).

We want to find the pair of solutions (x, y) such that:

Is minimum.

Note that our given equation is a line.

And the equation x²+y² is a circle centered on the origin.

In other words, we want to find the radius of the circle such that it is tangent to our line at 3x+y=15.

It must be tangent because this guarantees that it is the smallest value of  x²+y².

It's good if we have a visual of this. I've graphed the given linear equation. Please refer to it.

If you remember in geometry, in order for the radius to be tangent to a line, the radius must be perpendicular to our line.

So, let's find the perpendicular equation to our line. Our original equation is:

Subtract 3x from both sides:

So, the slope of our original equation is -3.

This means that the slope of our perpendicular line must be the negative reciprocal of -3. Namely, it is 1/3.

And since our circle is centered on the origin, this line will go through the origin. Therefore, our perpendicular equation is:

Graphing this will yield (please refer to the second graph):

Therefore, the intersection between our old and new line is at (4.5, 1.5).

Therefore, the (x, y) value that grants the minimum sum is 9/2 and 3/2.

We can check this by substituting them into our second equation. This yields:

Square and add:

Note that 22.5 is the radius squared.

Graphing this gives us (please refer to the third graph):

We can see that it is indeed tangent to our line. And it is the lowest value of P that does so.

So, our answer is x=9/2 and y=3/2.

boxplot e is the answer