Three Equation Rules

<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&nbsp;<strong>ANYTHING</strong>&nbsp;to both sides of the equation, as long as you&#39;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&nbsp;<em>simplify</em>&nbsp;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&nbsp;<em>almost</em>&nbsp;anything. However, there are two cases in which you have to be careful: <ul> <li><strong>Be Careful Case #1:&nbsp;</strong>You cannot divide each side by zero, as this would result in an &quot;undefined&quot; result. You&nbsp;<em>can</em>&nbsp;multiply both sides by $0$, but what&#39;s the point of that? Then you just get $0=0$.&nbsp;</li> <li><strong>Be Careful Case #2:&nbsp;</strong>If we&#39;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&#39;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>