An equation of the straight line which is perpendicular to 3x - 4y = 5 and passes through the point (-2, 1) is

Steps to Solve

1. Identify the slope of the given line

The first step is to rewrite the equation of the given line ( 3x - 4y = 5 ) in slope-intercept form ( y = mx + b ). This will help us find the slope.

Subtract ( 3x ) from both sides:

$$ -4y = -3x + 5 $$

Now, divide by (-4):

$$ y = \frac{3}{4}x - \frac{5}{4} $$

Thus, the slope ( m ) of the given line is ( \frac{3}{4} ).

2. Calculate the slope of the perpendicular line

For a line that is perpendicular, we take the negative reciprocal of the slope ( \frac{3}{4} ).

$$ m_{\text{perpendicular}} = -\frac{4}{3} $$

3. Use point-slope form to find the equation of the new line

We now use the point-slope form of the equation, ( y - y_1 = m(x - x_1) ), where the point ((-2, 1)) is given.

Substituting our values:

$$ y - 1 = -\frac{4}{3}(x + 2) $$

4. Simplify the equation

Distributing the slope:

$$ y - 1 = -\frac{4}{3}x - \frac{8}{3} $$

Add ( 1 ) (which is ( \frac{3}{3} ) in fractions) to both sides:

$$ y = -\frac{4}{3}x - \frac{8}{3} + \frac{3}{3} $$

Combine the constant terms:

$$ y = -\frac{4}{3}x - \frac{5}{3} $$

5. Convert to standard form

To convert into standard form ( Ax + By = C ):

Multiply through by ( 3 ) to eliminate fractions:

$$ 3y = -4x - 5 $$

Rearranging gives:

$$ 4x + 3y = -5 $$