Find the length of the line 2x+3y=6 intercepted between the axes and find the middle point of the intercepted portion. ans: `(3/2,1)`  

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lemjay | High School Teacher | (Level 3) Senior Educator

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First, let's solve for the intercepts of the equation.

To solve for the x-intercept, set y=0.

`2x + 3(0) =6`

`2x = 6`

`x = 3`

Hence, its x-intercept is (3,0).

To solve for the y-intercept, set x=0.

`2(0)+3y=6`

`3y=6`

`y=2`

Thus, the y-intercept is (0,2).

Then, plot these two points and connect them.

To determine the length of this line, let's consider its endpoints (3,0) and (0,2).

Then, apply the formula of distance between two points.

`d = sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`d=sqrt((0-3)^2+(2-0)^2)`

`d=sqrt(9 +4)`

`d=sqrt13`

Therefore, the length of the line 2x+3y=6 intercepted by the axes is `sqrt13` units.

To solve for the midpoint of (3,0) and (0,2), apply the formula:

`x_(mid) = (x_2+x_1)/2`

`y_(mid)=(y_2+y_1)/2`

Substituting the values to the formula, we would have:

`x_(mid)=(0+3)/2=3/2`

`y_(mid)=(2+0)/2=1`

Therefore, the middle point of the line 2x+3y=6 intercepted by the axes is `(3/2,1)` .

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