How do you find the equation of a line that passes through the point (2, 3) and is perpendicular to the line represented by the equation 2x + 3y = 6?

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To find the equation of the line that passes through the point (2, 3) and is perpendicular to the line given by the equation 2x + 3y = 6, we need to follow a series of steps:

Step 1: Determine the Slope of the Given Line

First, we need to rewrite the line equation in slope-intercept form, which is of the format y = mx + b, where m is the slope.

Starting with the equation:

2x + 3y = 6

Now, isolate y:

3y = -2x + 6
y = -\frac{2}{3}x + 2

From this, we can see that the slope of the line is m = -\frac{2}{3}.

Step 2: Find the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. Thus, the negative reciprocal of -\frac{2}{3} is:

m_{perpendicular} = \frac{3}{2}

Step 3: Use the Point-Slope Form of a Line

Next, we’ll use the point-slope form of the equation of a line, which is:

y - y_{1} = m(x - x_{1})

Here, (x_{1}, y_{1}) is the point through which the line passes, which in our case is (2, 3) and m is the slope we just calculated (\frac{3}{2}).

Substituting the values:

y - 3 = \frac{3}{2}(x - 2)

Step 4: Simplify to Get the Equation

Now, let’s simplify the equation:

y - 3 = \frac{3}{2}x - 3
y = \frac{3}{2}x

Finally, the equation of the line that passes through the point (2, 3) and is perpendicular to the line given by the equation 2x + 3y = 6 is:

y = \frac{3}{2}x

This line has a slope of \frac{3}{2} and passes through the point (2, 3).

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