[MathI] Manipulating Equations

Now that you know how to set up the equation, the next thing to do isto solve for the value that the question asks for. First and foremost,the most important thing to remember when manipulating equations is todo exactly the same thing to each side of the equation. If you divideone side of an equation by 3, you must divide the other side by 3. Ifyou take the square root of one side of an equation, take the squareroot of the other.
    By treating the two sides of the equationin the same way, you can rest easy that you won’t change the meaning ofthe equation. You will, of course, change the form of theequation—that’s the point of manipulating it. But the equation willalways remain true as long as you always do the same thing to bothsides.

    For example, let’s look at what happens when you manipulate the equation 3x + 2 = 5, with x = 1.

• Subtract 2 from both sides:

• Multiply both sides by 2:

• Add 4 to both sides:

    These examples show that you can tamper withthe equation in any way you want, as long as you commit the sametampering on both sides. If you follow this rule, you can manipulatethe question how you want without affecting the value of its variables.

    Solving an Equation with One Variable

    To solve an equation with one variable, youmust isolate that variable. Isolating a variable means manipulating theequation until the variable is the only thing remaining on one side ofthe equation. Then, by definition, that variable is equal to everythingon the other side, and you have successfully “solved for the variable.”

    For the quickest results, take the equationapart in the reverse order of operations. That is, first add andsubtract any extra terms on the same side as the variable. Then,multiply and divide anything on the same side of the variable. Next,raise both sides of the equation to a power or take their rootsaccording to any exponent attached to the variable. And finally, doanything inside parentheses. This process is PEMDAS in reverse(SADMEP!). The idea is to “undo” everything that is being done to thevariable so that it will be isolated in the end. Let’s look at anexample:

    In this equation, the variable x is being squared, multiplied by 3, added to 5, etc. We need to do the opposite of all these operations in order to isolate x and thus solve the equation.

    First, subtract 1 from both sides of the equation:

    Then, multiply both sides of the equation by 4:

    Next, divide both sides of the equation by 3:

    Now, subtract 5 from both sides of the equation:

    Again, divide both sides of the equation by 3:

    Finally, take the square root of each side of the equation:

    We have isolated x to show that x = ±5.

    Sometimes the variable that needs to beisolated is not conveniently located. For example, it might be in adenominator or an exponent. Equations like these are solved the sameway as any other equation, except that you may need differenttechniques to isolate the variable. Let’s look at a couple of examples:

Solve for x in the equation + 2 = 4.

    The key step is to multiply both sides by xto extract the variable from the denominator. It is not at all uncommonto have to move the variable from side to side in order to isolate it.

    Remember, performing an operation on avariable is mathematically no different than performing that operationon a constant or any other quantity.

    Here’s another, slightly more complicated example:

    This question is a good example of how it’snot always simple to isolate a variable. (Don’t worry about thelogarithm in this problem—we’ll review these later on in the chapter.)However, as you can see, even the thorniest problems can be solvedsystematically—as long as you have the right tools. In the nextsection, we’ll discuss factoring and distributing, two techniques thatwere used in this example.

    So, having just given you a very basic introduction to solving equations, we’ll reemphasize two things:

• Do the same thing to both sides.
• Work backward (with respect to the order of operations).

    Now we get into some more interesting tools you will need to solve certain equations.

    Distributing and Factoring

    Distributing and factoring are two of themost important techniques in algebra. They give you ways ofmanipulating expressions without changing the expression’s value. So itfollows that you can factor or distribute one side of the equationwithout doing the same for the other side of the equation.

    The basis for both techniques is the following property, called the distributive property:

    Similarly:

    a can be any kind of term, from a variable to a constant to a combination of the two.

    Distributing

    When you distribute a factor into anexpression within parentheses, you simply multiply each term inside theparentheses by the factor outside the parentheses. For example,consider the expression 3y(y2 – 6):

    If we set the original, undistributedexpression equal to another expression, you can see why distributingfacilitates the solving of some equations. Solving 3y (y2 – 6) = 3y3 + 36 looks quite difficult. But if you distribute the 3y, you get:

    Subtracting 3y3 from both sides gives us:

    Factoring

    Factoring an expression is essentially the opposite of distributing. Consider the expression 4x3 – 8x2 + 4x, for example. You can factor out the GCF of the terms, which is 4x:

    The expression simplifies further:

    See how useful these techniques are? You cangroup or ungroup quantities in an equation to make your calculationseasier. In the last example from the previous section on manipulatingequations, we distributed and factored to solve an equation. First, we distributed the quantity log 3 into the sum of x and 2 (on the right side of the equation). We later factored the term x out of the expression x log 2 – x log 3 (on the left side of the equation).

    Distributing eliminates parentheses, andfactoring creates them. It’s your job as a Math IC mathematician todecide which technique will best help you solve a problem.

    Let’s see a few examples:

    Combining Like Terms

    After factoring and distributing, there areadditional steps you can take to simplify expressions or equations.Combining like terms is one of the simpler techniques you can use, andinvolves adding or subtracting the coefficients of variables that areraised to the same power. For example, by combining like terms, theexpression:

    can be simplified to:

    by adding the coefficients of the variable x3 together and the coefficients of x2 together.

    Generally speaking, when you have anexpression in which one variable is raised to the same power indifferent terms, you can factor out the variable and add or subtractthe coefficients, combining them into one coefficient and thereforecombining the “like” terms into one term. A general formula forcombining like pairs looks something like this: