Finding Parallel Lines by Comparing Slopes
To find a line that is parallel to the given line, you need to ensure that it has the same slope. The given line is \(2y = 3x - 1\). We can rearrange this into slope-intercept form \(y = mx + b\) to find the slope \(m\).
Divide both sides of the equation by 2 to get the slope-intercept form:
\[ y = \frac{3}{2}x - \frac{1}{2} \]
Here, \(m = \frac{3}{2}\) is the slope.
Now we need to find the equation from the options that has the same slope of \(\frac{3}{2}\). Let's check each option by rewriting them in the slope-intercept form if necessary:
a. \(4y = 6x + 8\): Divide by 4 to get \(y = \frac{6}{4}x + 2\), which simplifies to \(y = \frac{3}{2}x + 2\). The slope here is \(\frac{3}{2}\), which is the same as the given line's slope. This line is thus parallel to the given line.
b. \(3y = 2x - 3\): Divide by 3 to get \(y = \frac{2}{3}x - 1\). The slope here is \(\frac{2}{3}\), which is not equal to \(\frac{3}{2}\).
c. \(2y = x - 3\): Divide by 2 to get \(y = \frac{1}{2}x - \frac{3}{2}\). The slope here is \(\frac{1}{2}\), which is not equal to \(\frac{3}{2}\).
d. \(y = \frac{1}{3}x - 1\): The slope here is \(\frac{1}{3}\), which is not equal to \(\frac{3}{2}\).
The correct answer is option a, as it's the only equation with a slope of \(\frac{3}{2}\), which means it is parallel to the given line.
