What is the equation of a line in the standard form passing through (4, -3) and perpendicular to 3x - 2y 5? Step-by-Step Solution

Finding the equation of a line that is perpendicular to another and passes through a specific point can be a straightforward process. Here’s a detailed step-by-step guide to derive the equation in standard form.

Step 1: Determine the Slope of the Given Line

To start, we need to rewrite the given equation, 3x - 2y 5, in slope-intercept form (y mx b) to find its slope.

Step 1.1: Rewrite the equation in slope-intercept form.

Starting with the original equation:

3x - 2y 5

Isolate y:

-2y -3x 5

Divide by -2:

y frac{3}{2}x - frac{5}{2}

From this, the slope m of the line is:

m frac{3}{2}

Step 2: Find the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another is the negative reciprocal of the other line's slope. Thus, the slope m_{perp} of the line we want to find is:

m_{perp} -frac{1}{m} -frac{1}{frac{3}{2}} frac{2}{3}

Step 3: Use the Point-Slope Form

Now that we have the slope of the line we want to find, we can use the point-slope form of the equation of a line (y - y1 m(x - x1)).

Given point: (x1, y1) (4, -3)

Equation: (y - (-3)) frac{2}{3}(x - 4)

Which simplifies to:

y 3 frac{2}{3}x - frac{8}{3}

Moving the -3 to the right side:

y frac{2}{3}x - frac{8}{3} - 3

Convert -3 to a fraction with a denominator of 3:

y frac{2}{3}x - frac{8}{3} - frac{9}{3}

Combine the constants:

y frac{2}{3}x - frac{17}{3}

Step 4: Convert to Standard Form

The standard form of a line is Ax By C. To convert from slope-intercept form (y mx b) to standard form, we can rearrange the equation.

-frac{2}{3}x - y -frac{17}{3}

To eliminate the fraction, multiply the entire equation by 3:

-2x - 3y -17

To express it in the standard form Ax By C with A being positive, multiply through by -1:

2x - 3y 17

Final Answer:

The equation of the line in standard form is: 2x - 3y 17

Conclusion

By following these steps, we have determined the equation of the line in standard form. The process involves determining the slope of the given line, finding the slope of the perpendicular line, using the point-slope form with the given point, simplifying, and finally converting to the standard form.