<p><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-three-equation-rules">Three Equation Rules Exercise</a></p><p><span style="color:#27ae60;">Three Equation Rules</span> can help us solve algebraic equations (isolate the variable):</p>
<p><strong>Rule 1</strong></p>
<ul>
<li>You can add or subtract <strong>ANYTHING</strong> to both sides of the equation, as long as you're adding or subtracting the same thing to both sides.
<ul>
<li>Of course, the goal of adding or subtracting something from both sides is to <em>simplify</em> the equation, i.e. get us closer to our goal of isolating the variable.</li>
</ul>
</li>
</ul>
<p><strong>Rule 2</strong></p>
<ul>
<li>You can multiply or divide both sides of the equation by <em>almost</em> anything. However, there are two cases in which you have to be careful:
<ul>
<li><strong>Be Careful Case #1: </strong>You cannot divide each side by zero, as this would result in an "undefined" result. You <em>can</em> multiply both sides by $0$, but what's the point of that? Then you just get $0=0$. </li>
<li><strong>Be Careful Case #2: </strong>If we're dividing by a variable (rather than a constant), some funny things can result.
<ul>
<li>$x^2=100x$
<ul>
<li>Notice here how we have two $x$ solutions: $0$ and $100$. If we were to divide both sides by $x$, we would get $x=100$. That's only one of our correct answers.</li>
</ul>
</li>
</ul>
</li>
</ul>
</li>
</ul>
<p><strong>Rule 3</strong></p>
<ul>
<li>When you have two equations involving the same variables, you can substitute information from one equation into the other. Check out the example below:</li>
</ul>
<p>$$y=x+2$$</p>
<p>$$x+3y=15$$</p>
<ul>
<li>We can substitute the $x+2$ in the first equation for the $y$ value in the second:</li>
</ul>
<p>$$x+3(x+2)=15$$</p>
<p>$$x+3x+6=15$$</p>
<p>$$4x=9$$</p>
<p>$$x=2.25$$</p>