What is the equation of the straight line parallel to `2x+3y+1=0` and passes through the point (-1, 2) ?

Options

A. `2x+3y-4=0`
B. `2x+3y-5=0`
C. `x+y-1=0`
D. `3x-2y+7=0`

Explanation

To find the equation of the straight line that is parallel to the line given by the equation \(2x + 3y + 1 = 0\) and passes through the point \((-1, 2)\), we can follow these steps:

### Step 1: Determine the slope of the given line
The equation of the line is given in the standard form \(Ax + By + C = 0\). Here, \(A = 2\), \(B = 3\), and \(C = 1\).

To find the slope \(m\) of the line, we can rearrange the equation into the slope-intercept form \(y = mx + b\):

\[ 3y = -2x - 1 \]
\[ y = -\frac{2}{3}x - \frac{1}{3} \]

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

### Step 2: Use the point-slope form of the equation of a line
Since we want to find a line that is parallel to the given line, it will have the same slope. We can use the point-slope form of the equation of a line, which is given by:

\[ y - y_0 = m(x - x_0) \]

Where \((x_0, y_0)\) is the point through which the line passes. In this case, \((x_0, y_0) = (-1, 2)\) and \(m = -\frac{2}{3}\).

Substituting these values into the point-slope form gives us:

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

### Step 3: Simplify the equation
Now, we will simplify the equation:

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

Adding 2 to both sides:

\[ y = -\frac{2}{3}x - \frac{2}{3} + 2 \]

To combine the constants, convert 2 into a fraction with a denominator of 3:

\[ y = -\frac{2}{3}x - \frac{2}{3} + \frac{6}{3} \]
\[ y = -\frac{2}{3}x + \frac{4}{3} \]

### Step 4: Convert to standard form
To convert this equation into standard form \(Ax + By + C = 0\), we can rearrange it:

\[ \frac{2}{3}x + y - \frac{4}{3} = 0 \]

Multiplying through by 3 to eliminate the fraction:

\[ 2x + 3y - 4 = 0 \]

### Final Equation
Thus, the equation of the straight line that is parallel to \(2x + 3y + 1 = 0\) and passes through the point \((-1, 2)\) is:

\[ 2x + 3y - 4 = 0 \]