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# Algebra

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Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

Jul 8, 2024

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Let's note something important before we begin.

All the points 5 units away from (2, 7) forms a circle with radius 5 and center of (2, 7).

A circle with that equation would look something like

$$(x-2)^2+(y-7)^2 = 5^2$$

Combining that with the second part of the problem, we have a system with

$$(x-2)^2+(y-7)^2 = 25\\y=5x-28$$

Now, we already have a value for y, and we subsitute that into the first equation to isolate x.

We find that

$$(x-2)^2+(5x-35)^2 = 25$$

Expanding and bringing all terms to one side of the equation, we get a quadratic. We find that

$$26x^2-354x+1204=0$$

This gives us 2 x values that will satisfy thsi equation. We get that

$$x=7,\:x=\frac{86}{13}$$

Now, we plug these two values of x back into the equation to find y, so that we can complete our problem.

We find that $$y = 7, y= \frac{66}{13}$$

So our final two abswers that satisfy all these conditions given above in the problem are

$$(7, 7), (86/13, 66/13)$$

Thanks! :)

Jul 8, 2024