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Variable Equations

Plugging in Numbers (Test Prep Algebra!)

 

Sometimes the secret to solving starndardized test algebra problems is just plugging in numbers.  Often the problem will give some parameters for the numbers to plug in (e.g., they must be positive or negative).  Other times, the most efficient way to solve a problem is to plug in the answer choices.  As long as you know the rules of math and algebra, you can plug in numbers and see what your answers come out to!

There are two main types of plugging in questions.  

Solving for a Variable in an Exponent

Sometimes algebra problems will put the variable that you have to solve for in an exponent.  These problems can look intimidating but work exactly the same as other “plug in” problems.  Just find the number that makes the exponent equation true!

Getting the vocabulary right makes it easier to talk about exponent problems so:

Using Special Symbols

Sometimes SAT writes equations using symbols that aren’t actually math symbols.  They use stars or hearts or other strange symbols.  The symbols have no intrinsic meaning; you just do what the problem tells you to do by following the example. 

Sometimes special symbols are just “blanks.”  You just have to replace the symbol with a number that works. 

Example:

           $\text{If }6+\bigtriangleup=18\text{, what is }\bigtriangleup\text{?}$

Functions

A function is a relationship between two variables.  Usually written as $f(x)$ or $g(x)$, functions are just equations (like $y = x– 4$). For example, $f(x)= x - 4$ means that for every value of $x$, $f(x)$ is $x - 4$ or $y=x-4$.

Functions tend to scare students.  For many students, as soon as they see $f(x)$, they assume that they cannot do a problem.  But functions just show how two variables are related to each other.

Solve "In Terms of" a Given Variable

Many equations contain several variables, or several unknowns. Oftentimes, we are given or can solve for the value of one or more variables and use those values to solve for the values of other variables. However, other times, you don't have a value -- or don't want one!  You just want to rearrange an equation to solve for one variable "in terms of" another variable. 

The phrase "in terms of" just means that when you solve for a variable, that "in terms of" variable is going to appear in your answer.

Substitute from Word Problems

You already know that in algebra, variables represent values, and if you know the value, you can substitute the value for the variable in an equation.

The rule of substitution (any values that are equal can be freely substituted for each other) also holds true for word problems.

Many word problems give you an equation and a value for a variable.  All you have to do is plug in the value for the variable and solve.

Let's try one of those word problems.

Example:

Using Balance

The primary rule when solving an algebra equation is: you can do whatever you want to one side of an equation as long as you do the same thing to the other side.  So, you can add something to both sides of an equation, multiply or divide both sides of an equation by the same number, square or square root both sides of an equation.  Every operation is fair game as long as you perform the same operation on the other side of the equation to keep the equation in balance.

Distributive Property

The distributive property says that if you multiply a term times multiple other terms (this usually looks like a term multiplied by a polynomial in parentheses) you multiply the first term by each term in the parentheses, and then combine the products. Essentially, you get rid of the parentheses by "distributing" the multiplier term to each term inside the parentheses.

Combining Like Terms and Solving

When you get into long and complicated algebra equations, there are often multiple ways to solve them. 

One of the first things you should always do when working with equations, is combine like terms.  Combining like terms will not only make your equation shorter and easier to work with, but it will cut down on the number of steps you have to do when isolating a variable.

Equations with Fractions

Solving an equation with fractions is the same as solving an equation without fractions.  The exact same rules apply: isolate the variables by using "opposite" operations to remove (or "undo") the terms that are attached to the variable.  "Undo" terms in reverse order of operations  (so, undo operations attached with addition and subtraction, then multiplication and division).