Basic Algebra Study Guide for the Math Basics

How to Prepare for the Questions Involving Basic Algebra on a Math Test

General Information

The ancient Greeks laid the foundations of mathematical reasoning using logic and geometry to explain complex patterns and relationships. About a thousand years later, Persia had its own “renaissance” and a man named Muhammad Al-Kharizmi began developing a new form of math known as al-jabr—what we know today as algebra. Its principles lie on using symbols (variables) to represent unknown quantities in equations. Today, in addition to the logic of the Greeks, algebra is the standard language of mathematics used in calculus, statistics, and even geometry, to name a few.

Expressions, Equations, and Inequalities

In contrast to geometry (the study of shapes), algebra uses a combination of numbers and symbols to make up expressions, equations, and inequalities. You have to know some of the basic symbols when dealing with algebra.

\[\begin{array}{|c|c|} \hline \textbf{Symbol} & \textbf{Meaning} \\ \hline + & \text{plus, addition, positive} \\ \hline - & \text{minus, subtraction, negative} \\ \hline \times \text{ or } \cdot & \text{times, multiply} \\ \hline \div \text{ or } / & \text{divide by, division} \\ \hline = & \text{equals} \\ \hline \text{^ or } ^2 & \text{to the power of, exponent} \\ \hline < & \text{less than} \\ \hline > & \text{greater than} \\ \hline \end{array}\]

Expressions

An expression is a combination of numbers and/or variables without an equality sign. Think of it this way: a mathematical expression can be viewed as an english phrase. It’s not the whole sentence, just part of it. Here are some examples of expressions:

\[5\] \[2x+7\] \[\dfrac{45-x^2}{2y}\]

When you read them, you’ll find they don’t make a whole sentence: there is no subject, verb, and direct object. The number \(5\) is just “five”. There are more parts here, but \(2x+7\) is just “two x plus 7”. And \(\dfrac{45-x^2}{2y}\) is just “the difference of 45 and x squared divided by two y.”

When confronted with expressions, there isn’t much you can do to them. The directions to a problem will only say to simplify.

There are certain algebraic properties you can use when simplifying.

\[\begin{array}{|c|c|} \hline \textbf{Property} & \textbf{Expression} \\ \hline \text{distributive} & 2(x+5) = 2x + 10 \\ \hline \text{commutative (addition or multiplication)} & 3+2=2+3 \text{ or } 5\cdot x = x\cdot5 \\ \hline \text{associative (addition or multiplication)}& 3+(2+6)=(3+2)+6 \text{ or } 5 \cdot (6 \cdot x) = (5 \cdot 6) \cdot x \\ \hline \end{array}\]

Equations

Once two expressions are compared using an equals sign, the sentence is complete and an equation is formed. For example, the equation \(2x+7 = 13\) is a full sentence “two x plus 7 is 13”. When faced with an equation, the directions will usually be to solve the equation. To do this, isolate the variable by performing a series of operations to both sides of the equation at the same time, keeping the equation balanced. This is the basic truth of algebra: “Whatever you do to one side of the equation, you must do to the other side.” You’ll use these properties of algebraic equalities when solving.

\[\begin{array}{|c|c|} \hline \textbf{Property} & \textbf{Expression} \\ \hline \text{reflexive} & a=a\\ \hline \text{symmetric} & \text{If } x = 3\cdot 2 \text{, then } 3\cdot 2 = x \\ \hline \text{transitive} & \text{If } a=b \text{ and } b=c \text{, then } a=c \\ \hline \text{substitution} & \text{If } a=b \text{, then } a \text{ may be replaced by } b \text{ in any expression.}\\ \hline \text{addition prop of equality} & \text{If } a= b \text{, then } a+c=b+c \\ \hline \text{subtraction prop of equality} & \text{If } a= b \text{, then } a-c=b-c \\ \hline \text{multiplication prop of equality} & \text{If } a= b \text{, then } a\cdot c=b\cdot c \\ \hline \text{division prop of equality} & \text{If } a= b \text{, then } a\div c=b\div c \\ \hline \end{array}\]

Inequalities

Inequalities are formed when two expressions are compared with an inequality sign instead of an equals sign.

Common inequality symbols:

\[\begin{array}{|c|c|} \hline \textbf{symbol} & \textbf{meaning} \\ \hline \lt & \text{less than} \\ \hline \gt & \text{greater than} \\ \hline \le & \text{less than or equal to, at most} \\ \hline \ge & \text{greater than or equal to, at least} \\ \hline \ne & \text{not equal to} \\ \hline \end{array}\]

Just like an equation, inequalities are solved. However, there is one major difference.

When multiplying or dividing both sides of an inequality by a negative value, you must flip the inequality sign

Let’s see why this makes sense. You know that \(3 \gt -1\). But let’s multiply both sides of the equation by \(-1\). Now, \(-3 \gt 1\) is not true, but \(-3 \lt 1\) is.

Algebraic Concepts

Equations and expressions are split into things called terms: numbers, variables, or products of numbers or variables. Terms are separated from each other by addition or subtraction.

Variable

A variable is just a letter or symbol used to represent an unknown quantity. Usually, we use letters like x, y, or a, among other letters. Sometimes we use greek letters like \(\alpha, \; \gamma, \; \text{or } \theta\) (alpha, gamma, or theta) when writing about unknown angles. We use the word variable because the unknown value can change (or vary) depending on the rest of the expression or equation.

Constant

A constant is any number by itself, not connected to a variable. It’s called a constant because its value doesn’t change, like a variable. In the expression \(4x^2 + 5x +6\), the constant term is \(6\).

Coefficient

Occasionally, variables have multipliers assigned to them, like \(5x\) or \(-3z\). These multipliers are called coefficients. So, \(5\) is the coefficient of \(5x\) and \(-3\) is the coefficient of \(-3z\). Sometimes, a variable stands alone, like \(y\), but in that case, the coefficient is \(1\) because \(y\) is the same as \(1y\).

Like Terms

An expression might contain like terms: terms that have the same variable(s) with the same exponent. To combine like terms, simply add or subtract the coefficients. So, you can combine any terms with the same variable, like \(2x\) and \(4x\), being sure to notice the sign in front of each. But, you cannot combine \(2x\) and \(4y\), because they do not have the same variable. Here’s an example:

Simplify: \(4x+2y-3x\)

In this case, \(4x\) and \(-3x\) are like terms (same variable, no exponent to worry about). Let’s use the commutative property to move some things around:

\[4x -3x + 2y\]

Now, combine the like terms: \(4-3=1\) so \(4x-3x=1x\)

\[1x+2y\]

or

\[x+2y\]

Here’s a more involved example to demonstrate a deeper understanding of combining like terms.

Simplify: \(12x^2 -3xy + 4x - 2y^2 +5x^2 - 5xy +4xy^2 + 2y^2\)

First, use the commutative and associative properties to group like terms:

\[(12x^2 + 5x^2) + (-3xy-5xy) + 4x + (-2y^2 + 2y^2) + 4xy^2\]

Then, combine those like terms:

\[17x^2 + -8xy + 4x + 0y^2 + 4xy^2\] \[17x^2 - 8xy + 4x + 4xy^2\]

Note, \(-8xy\) and \(4xy^2\) are not like terms because, while they have the same variables, each variable doesn’t have the same exponent.