<p><span style="color:#27ae60;">Parallel Lines</span> are two lines that never touch each, like an unhappily married couple. In coordinate geometry, you know that two lines are parallel if they have the <u>same slope</u>.</p>
<p><strong><span style="color:#8e44ad;">Example</span></strong></p>
<p>Let's imagine we have a line with the equation below:</p>
<p>$$6x-2y=10$$</p>
<p>Let's convert it into the "very important equation."</p>
<p>$$-2y=-6x+10$$</p>
<p>$$y=3x-5$$</p>
<p>In this form, it's obvious the slope is $3$. So here's the thing. EVERY line with a slope of $3$ is parallel to this bad boy. Here are some examples:</p>
<p style="text-align: center;"><strong><span style="color:#8e44ad;">Parallel to $y=3x-5$</span></strong></p>
<p>$$y=3x+7$$</p>
<p>$$y=3x-100$$</p>
<p>$$y=3x+100!$$</p>
<p><strong><span style="color:#e74c3c;">What about two congruent equations? Are they parallel?</span></strong></p>
<p>Good question. Let's say we have $y = x$ and $2y =2x$, two separate lines. Well, they still have the same slope. In fact, they're the SAME equation, just in a different form. Are the lines parallel? Technically yes. Will GRE ask about this? Likely not.</p>