calculate the shortest distance between the given line and point 2x + 3y = 6; (7, 6) 

To calculate the shortest distance between a given line and a point, we can use the following formula:

\[
\text{Distance} = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

Where:
- \(Ax + By + C = 0\) is the equation of the line.
- \((x_1, y_1)\) is the given point.

In this case, the equation of the line is \(2x + 3y = 6\), which can be rewritten in the standard form:

\[
2x + 3y - 6 = 0
\]

Here:
- \(A = 2\)
- \(B = 3\)
- \(C = -6\)

The given point is \((x_1, y_1) = (7, 6)\).

Now, substituting these values into the distance formula:

\[
\text{Distance} = \frac{|2(7) + 3(6) - 6|}{\sqrt{2^2 + 3^2}}
\]

\[
\text{Distance} = \frac{|14 + 18 - 6|}{\sqrt{4 + 9}}
\]

\[
\text{Distance} = \frac{|26|}{\sqrt{13}} = \frac{26}{\sqrt{13}}
\]

To simplify further:

\[
\text{Distance} = \frac{26}{\sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{26\sqrt{13}}{13} = 2\sqrt{13}
\]

So, the shortest distance between the point \((7, 6)\) and the line \(2x + 3y = 6\) is \(2\sqrt{13}\) units.

Would you like more details or have any questions?

### Related Questions:
1. How do you find the distance between two parallel lines?
2. How do you derive the distance formula used between a point and a line?
3. How would the formula change if the line was in 3D space?
4. What are the conditions for a point to lie on the line itself?
5. How do you find the intersection point between two lines?

**Tip:** Always make sure to convert the equation of the line into the standard form \(Ax + By + C = 0\) before applying the distance formula.

Calculate the shortest distance between the given line and point 2x + 3y = 6; (7, 6)

Find the shortest distance between the line 2x + 3y = 6 and the point (7, 6) using the distance formula. This detailed solution explains the key steps and mathematical concepts involved, perfect for high school students in Grades 9-12.

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