Finding the Equation of a Parallel Line

To find the equation of a straight line that is parallel to a given line and passes through a specific point, we need to understand the relationship between parallel lines and their slopes.

Understanding Parallel Lines and Slope

Two distinct lines are parallel if and only if they have the same slope. The equation of a straight line is often given in the form \( Ax + By + C = 0 \). The slope (\(m\)) of a line in this form is given by the formula \( m = -\frac{A}{B} \).

The given line is \( 2x + 3y + 1 = 0 \). Comparing this to \( Ax + By + C = 0 \), we have \( A = 2 \) and \( B = 3 \). The slope of this given line is:

\( m_{\text{given}} = -\frac{A}{B} = -\frac{2}{3} \)

Since the line we are looking for is parallel to \( 2x + 3y + 1 = 0 \), it must have the same slope. So, the slope of the required line is \( m_{\text{required}} = -\frac{2}{3} \).

General Form of the Parallel Line Equation

A line parallel to \( 2x + 3y + 1 = 0 \) will have an equation of the form \( 2x + 3y + k = 0 \), where \( k \) is a constant that we need to determine.

Using the Given Point to Find the Constant \( k \)

We are told that the required straight line passes through the point \( (-1, 2) \). This means that the coordinates \( x = -1 \) and \( y = 2 \) must satisfy the equation of the line \( 2x + 3y + k = 0 \). We can substitute these values into the equation to solve for \( k \):

\( 2(-1) + 3(2) + k = 0 \)

Now, let's simplify and solve for \( k \):

\( -2 + 6 + k = 0 \)

\( 4 + k = 0 \)

\( k = -4 \)

Writing the Final Equation of the Straight Line

Now that we have found the value of \( k \), we can substitute it back into the general form of the parallel line equation \( 2x + 3y + k = 0 \):

\( 2x + 3y + (-4) = 0 \)

\( 2x + 3y - 4 = 0 \)

This is the equation of the straight line that is parallel to \( 2x + 3y + 1 = 0 \) and passes through the point \( (-1, 2) \).

Comparing with Options

Let's look at the given options:

• Option 1: \( 2x + 3y - 4 = 0 \)
• Option 2: \( 2x + 3y - 5 = 0 \)
• Option 3: \( x + y - 1 = 0 \)
• Option 4: \( 3x - 2y + 7 = 0 \)

Our calculated equation \( 2x + 3y - 4 = 0 \) matches Option 1.

Summary of Steps

1. Identify the slope of the given line \( 2x + 3y + 1 = 0 \).
2. Use the fact that parallel lines have the same slope.
3. Write the general form of the equation for a line with this slope (\( 2x + 3y + k = 0 \)).
4. Substitute the coordinates of the given point \( (-1, 2) \) into the general equation.
5. Solve for the constant \( k \).
6. Substitute the value of \( k \) back into the general equation to get the final equation.

Revision Table: Straight Lines and Parallelism

Additional Information: Exploring Line Equations

The form \( Ax + By + C = 0 \) is called the standard form of a linear equation. Another useful form is the slope-intercept form, \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept (the point where the line crosses the y-axis).

If we had the slope \( m = -\frac{2}{3} \) and the point \( (-1, 2) \), we could also use the point-slope form: \( y - y_1 = m(x - x_1) \). Substituting the values:

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

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

To convert this to the standard form, multiply by 3:

\( 3(y - 2) = -2(x + 1) \)

\( 3y - 6 = -2x - 2 \)

Move all terms to one side:

\( 2x + 3y - 6 + 2 = 0 \)

\( 2x + 3y - 4 = 0 \)

This confirms our result obtained using the general parallel line form. Both methods yield the same equation for the straight line.