Find the equations of the lines passing through the point (4,5) parallel to lines `3x=4y+7 `.

Answer

To find the equation of the line passing through the point (4, 5) and parallel to the line given by the equation \(3x = 4y + 7\), we can follow these steps:

### Step 1: Rewrite the given line in standard form
We start with the equation \(3x = 4y + 7\). We can rearrange this into the standard form of a line, which is \(Ax + By + C = 0\).

\[ 3x - 4y - 7 = 0 \]

### Step 2: Identify the slope of the given line
From the standard form \(3x - 4y - 7 = 0\), we can determine the slope of the line. The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope.

To find the slope, we can rearrange the equation:

\[ 4y = 3x - 7 \implies y = \frac{3}{4}x - \frac{7}{4} \]

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

### Step 3: Use the point-slope form to find the equation of the new line
Since we want to find a line parallel to the given line, it will have the same slope \(\frac{3}{4}\). We can use the point-slope form of the line equation, which is given by:

\[ y - y_1 = m(x - x_1) \]

Here, \((x_1, y_1) = (4, 5)\) and \(m = \frac{3}{4}\).

Substituting these values into the point-slope form:

\[ y - 5 = \frac{3}{4}(x - 4) \]

### Step 4: Simplify the equation
Now we will simplify this equation:

\[ y - 5 = \frac{3}{4}x - 3 \]

Adding 5 to both sides:

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

### Step 5: Convert to standard form
To convert this into standard form \(Ax + By + C = 0\), we can rearrange it:

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

Multiplying through by 4 to eliminate the fraction:

\[ -3x + 4y - 8 = 0 \]

Rearranging gives us:

\[ 3x - 4y + 8 = 0 \]

### Final Answer
The equation of the line passing through the point (4, 5) and parallel to the line \(3x = 4y + 7\) is:

\[ 3x - 4y + 8 = 0 \]