Find the equation of the line parallel to the line \(3x - 4y = 12\) and passing through the point \((2, -3)\).
Answer
Step 4: Simplifying the equation, we get: \[y + 3 = \frac{3}{4}x - \frac{3}{2}\] \[y = \frac{3}{4}x - \frac{3}{2} - 3\] \[y = \frac{3}{4}x - \frac{9}{2}\]
Step 1 ：Step 1: Convert the given line to slope-intercept form (\(y = mx + b\)), where \(m\) is the slope. So, we have: \[3x - 4y = 12 \Rightarrow y = \frac{3}{4}x - 3\] The slope of this line is \(\frac{3}{4}\).
Step 2 ：Step 2: A line parallel to another line will have the same slope. So, the slope of the line we are trying to find is also \(\frac{3}{4}\).
Step 3 ：Step 3: We know the slope (\(m\)) and a point on the line (\((2, -3)\)). We can plug these values into the point-slope form of a line equation: \[y - y_{1} = m(x - x_{1})\] where \((x_{1}, y_{1})\) are the coordinates of the known point. Substituting for \(m\), \(x_{1}\), and \(y_{1}\), we get: \[y - (-3) = \frac{3}{4}(x - 2)\]
Step 4 ：Step 4: Simplifying the equation, we get: \[y + 3 = \frac{3}{4}x - \frac{3}{2}\] \[y = \frac{3}{4}x - \frac{3}{2} - 3\] \[y = \frac{3}{4}x - \frac{9}{2}\]
