Find the equation of the line parallel to 2 x - 3y + 5 = 0 and passing through (1,1)

Options

A. 2 x - 3y + 1 = 0
B. 2x + 3y + 1 = 0
C. 3 x - 2y + 1 = 0
D. None of these

Answer

To find the equation of the line that is parallel to the line given by the equation \(2x - 3y + 5 = 0\) and passes through the point \((1, 1)\), follow these steps:

### Step 1: Identify the slope of the given line
The given line is in the form \(Ax + By + C = 0\). We can rewrite it in slope-intercept form \(y = mx + b\) to find the slope \(m\).

Starting with:
\[ 2x - 3y + 5 = 0 \]
Rearranging gives:
\[ -3y = -2x - 5 \]
Dividing by -3:
\[ y = \frac{2}{3}x + \frac{5}{3} \]
Thus, the slope \(m\) of the given line is \(\frac{2}{3}\).

### Step 2: Use the point-slope form to find the equation of the parallel line
Since parallel lines have the same slope, the slope of the line we are looking for is also \(\frac{2}{3}\). We can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point through which the line passes, and \(m\) is the slope.

Substituting \(m = \frac{2}{3}\) and the point \((1, 1)\):
\[ y - 1 = \frac{2}{3}(x - 1) \]

### Step 3: Simplify the equation
Now we simplify the equation:
\[ y - 1 = \frac{2}{3}x - \frac{2}{3} \]
Adding 1 to both sides:
\[ y = \frac{2}{3}x - \frac{2}{3} + 1 \]
Converting 1 to a fraction with a denominator of 3:
\[ y = \frac{2}{3}x - \frac{2}{3} + \frac{3}{3} \]
Combining the constants:
\[ y = \frac{2}{3}x + \frac{1}{3} \]

### Step 4: Convert to standard form
To convert this back to standard form \(Ax + By + C = 0\):
\[ y - \frac{2}{3}x - \frac{1}{3} = 0 \]
Multiplying through by 3 to eliminate the fractions:
\[ 3y - 2x - 1 = 0 \]
Rearranging gives:
\[ 2x - 3y + 1 = 0 \]

### Final Answer
Thus, the equation of the line parallel to \(2x - 3y + 5 = 0\) and passing through the point \((1, 1)\) is:
\[ 2x - 3y + 1 = 0 \]

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