Shortest distance between Y=-3x+2 and point U(7,3).

Calculate the shortest distance between each point and the given line, Please show a full explanation and graph, thank you.

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To find the shortest distance between U and the line, we need to have a line from U to the line y=-3x+2 that is perpendicular to y=-3x+2. If we call the intersection point between these two lines Q, then since y=-3x+2 has slope -3, the line UQ has slope `1/3` . By using U, we get the equation of UQ:

`y=mx+b`

`3=1/3(7)+b`

`3-7/3=b`

`b=2/3`

So the line UQ is `y=1/3x+2/3` .

The intersection between UQ and y=-3x+2 is found by equating the lines:

`1/3x+2/3=-3x+2` multiply by 3

`x+2=-9x+6` collect like terms

`x+9x=6-2` simplify

`10x=4`

`x=4/10`

`x=2/5`

Sub into y=-3x+2 to get:

`y=-3(2/5)+2`

`=-6/5+10/5`

`=4/5`

So the point Q is `(2/5,4/5)` . The shortest distance is from U to Q. This is found through the distance formula:

`d=sqrt{(7-2/5)^2+(3-4/5)^2}`

`=sqrt{33^2/5^2+11^2/5^2}`

`=sqrt1210/5`

`={11sqrt10}/5`

**The shortest distance is `{11sqrt10}/5` . The graph is:**

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