Question - Finding Parallel Lines by Comparing Slopes
Solution:
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.
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