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Algebra I

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).

Solve with Given Values

When you first look at it, algebra looks like math with letters.

But, what you quickly learn is that those letters, called variables, represent numbers.  You just don't know what numbers they represent yet!

In advanced algebra problems, you will solve for these variables (or the lines created by these variables).

But, at the most basic level, you can always replace variables in equations with the numbers that those variables stand for. 

Solve for a Variable

Algebra is when, in math, you use variables in place of numbers that you don't know.  Those variables stand for a number, you just don't know what it is.

When you are given an algebraic equation with just one variable, you have all of the information you need to figure out the value of that variable.

Think about this example:

$x + 5 = 9$

Do you know what $x$ equals?

Sure you do.  You know that $4+5=9$ so $x$ must equal $4$.

Creating Expressions, Equations, and Inequalities

One of the most critical skills in algebra is learning to write an equation.  Being able to translate a problem -- whether a word problem or a real life problem -- into an equation opens up an entire realm of math.

So, how do you translate words into an equation?

 

First, some basics: 

Creating Equations (Unknowns in terms of same variable)

In algebra we use variables to represent unknowns.  And, by using the tips and skills in the Creating Equations lesson, we can put together an equation from most word problems.  However, it's difficult (and sometimes impossible) to solve a single equation with two variables.  So, how can we take an equation with two unknowns and write an equation with only one variable?

Often word problems tell you how the two unknowns are related to each other, giving you a way to write one uknown "in terms of" another unknown.