Study Guide: Equations
Vocabulary
The following terms are important to understand when working with algebraic equations.
| Term | Definition |
|---|---|
| Mathematical Term | A mathematical expression that does not contain +, - or equality symbols. |
| Coefficient | The number being multiplied to a variable in a mathematical term. |
| Constant Term | A mathematical term with no variable or exponents. |
| Like Terms | Two or more mathematical terms having the same variable and the same exponent. |
| Equation | Two mathematical expressions that have been set equal to each other, such as $2 + 5 = 7$. |
| Equivalent Equations | Two equations that look different, but are mathematically identical. |
| Inequality | An equation in which the two sides are not equal. |
| Variable | A symbol (or placeholder) for a value we don’t yet know. |
| Exponent | The number of times the base of a power should be multiplied. |
| Literal Equation | Equations made up of alphabet variables, such as $E = MC^2$. |
| Inverse Operation | An operation that reverses another operation, such as addition and subtraction. |
| Ratio | TWo values that have been set in relation to each other. |
| Proportion | Two ratios that are set in relation to each other. |
| Rate | A proportion in which one value has been set to 1. |
| Cross Multiplication (or Cross Product) | Multiplying the numerators of fraction with each other’s denominators. |
| General Mathematical Principle | A core principle of mathematics. |
| Addition Property of Equality | If we add or subtract the same number to both sides of an equation, the sides remain equal. |
| Multiplication Property of Equality | If we multiply or divide the same number to both sides of an equation, the sides remain equal. |
| Commutative Property of Addition | The order in which numbers are added does not change the sum. (This property does not apply to subtraction.) |
| Commutative Property of Multiplication | The order in which numbers are multiplied does not change the sum. (This property does not apply to division.) |
| Associative Property of Addition | The order in which numbers are grouped when adding does not change the sum. (This property does not apply to subtraction.) |
| Associative Property of Multiplication | The order in which numbers are grouped when multiplying does not change the product. (This property does not apply to division.) |
| Distributive Property | |
| Identity Property of Addition | When a number is added to zero, the result is the original number. |
| Identity Property of Multiplication | When a number is multiplied by 1, the result is the original number. |
| Justification | The mathematical principle used to transform an equation into another form. |
| Solution | The answer to a mathematical problem. |
| Solution Set | The set of all possible solutions to a mathematical problem. |
About
Logical reasoning is central to all mathematics, and is at the heart of algebraic thinking. A proper justification for each step in the solution of an equation is not simply a description of what was done (“I added 4 to both sides.”), but rather a general principle, such as The Addition Property of Equality.
A solution is a value for a variable that makes an equation true. it is important to understand that not all equations have a single solution. Some have many or even infinite solutions. Some have no solutions.
Solving equations is one of the most important skills in mathematics (and science, medicine, engineering, economics, computer programming, etc.).
Algebra is a technique for transforming an equation into other equivalent equations that are easier to understand. During the solution process, it must be possible to justified each transformation by a general mathematical principle.
Solving Equations
We solve equations by transforming them into equivalent equations that are easier to understand. By moving a variable to one side of the equal sign and everything else to the other side, we can more easily see what the variable equals.
For example, in the following equation, we use the Addition Property of Equality by subtracting 3 from both sides to isolate the variable. (Note: Subtraction is a form of addition. For example: $8 - 3 = 8 + (-3) = 5$
\begin{align} x + 3 &= 8 &&\text{Given} \tag 1 \\[2ex] x + 3 {\color{red}-3} &= 8 {\color{red}- 3} &&\text{Justification: Addition Property of Equality} \tag 2 \\[2ex] x &= 5 &&\text{Solution (All three equations are equivalent.)} \tag 3 \\[2ex] \end{align}
Combining Like Terms
One reason algebra seems difficult is that there are many ways to simplify an equation. It can be hard to decide which method to use. One important skill is combining like terms. This is one if the easiest ways to begin simplifying an equation. For example, we know we can combine $3$ and $5$ to get $8$.
$$3+5=8$$
This works because $3$ and $5$ are “like terms”. That is, they are both constant (unchanging) values, and do not include variables or exponents.
That much was easy, but what happens when we do have variables and exponents? To combine variable and exponential terms correctly, we need to understand when they can be combined.
Term A term is a mathematical expression that DOES NOT include plus (+), minus (-) or equality symbols (=, <, >, etc.).
Like Terms: The big deal about like terms is that they can be combined. This fact helps us simplify equations.
There are constant terms, variable terms, and exponential terms. Just as the constant terms $2$ and $3$ can be combined (added) to make $5$, variable and exponential terms can also be combined.
For example, we can combine 2 apples and 3 apples to get 5 apples, but we sure can’t combine 2 apples and 3 oranges to get 5 apples. That would be combining unlike variables. Apples and oranges are not alike enough to be combined in that way.
Constant Terms: Terms with no variable or exponents are called Constant Terms. They usually have an implied variable. Think about the above apples example. We can just say $2 + 3 = 5$ without showing that we are talking about apples, but that really only makes sense because in the back of our minds we know we’re counting apples.
This equation can be written many different ways, some of which show the implied variable.
\begin{align} 2 + 3 &= 5 \\\ 2 \text { apples } + 3 \text{ apples } &= 5 \text{ apples} \\\ \\\ \hline \text{Let } a &= \text{ apples} \\\ \hline 2a + 3a &= 5a \\\ \end{align}
So… here’s the definition of Like Terms: Like terms have the same variable and the same exponent.
Like and Unlike Terms
| Like Terms | Type |
|---|---|
| $3 + 8.5$ | These two constant terms can be combined. |
| $\frac{1}{8}{\color{teal}x} + 14{\color{teal}x}$ | Like variable terms |
| $3x^{\color{teal}2} + 4x^{\color{teal}2}$ | Like exponential terms |
| Unlike Terms | Type |
|---|---|
| $3 + 8.5{\color{red}x}$ | Can’t combine constant and variable terms! |
| $\frac{1}{8}{\color{red}x} + 14{\color{red}y}$ | Can’t combine different variables! |
| $x^{\color{red}2} + x^{\color{red}4}$ | Can’t combine different exponents! |
What is a Coefficient?
A coefficient is a fancy word for the number being multiplied to a variable. For example, in the expression $5x$, (which means $5 \times x$), the coefficient is $5$ and the variable is $x$.
Careful! Variable expressions with no coefficient have an invisible coefficient of $1$. For example, the expression ($x$) really means ($1x$), or ($1 \times x$). We do this because mathematicians are lazy, and don’t want to write $1$ all the time.
In the same way, the number $4$ really means $\frac{4}{1}$, or “four ones”. If you doubt this, test it by putting any other number in the denominator and see if the fraction still equals $4$.
Simplifying Equations
Simplify: \( 6x + 3 = x + 8 \)
Simplifying Equations Containing Fractions
Simplify: \( \dfrac{3p}{8} + \dfrac{7p}{16} - \dfrac{3}{4} = \dfrac{1}{4} + \dfrac{p}{16} + \dfrac{1}{2} \)
- Remove the denominators by multiplying every term on both sides by the Least Common Denominator (LCD).
- Combine like terms.
- Isolate the variable.
Solving Word Problems with One Variable
A phone company charges $0.027 per minute and a $2 monthly fee. Another phone company charges $0.035 per minute with no monthly fee.
- Find the number of minutes at which the charges for both companies are the same.
- What is that cost?
- Let \( m \) = number of minutes
- Company A: \( 2 + 0.027m \)
- Company B: \( 0.035m \)
- Set the expressions equal to each other and solve for $m$: \( 2 + 0.027m = 0.035m \)
Find the number of minutes.
- Let \( m \) = number of minutes
- Company A: \( 2 + 0.027m \)
- Company B: \( 0.035m \)
- Set the expressions equal to each other and solve for $m$.
\begin{align} 2 + 0.027m &= 0.035m &&\text{Given. Multiply all terms by 16.} \tag 1 \\[2ex] 2 &= 0.008m &&\text{Divide both sides by 0.008.} \tag 2 \\[2ex] 250 &= m &&\text{Solution.} \tag 3 \\[2ex] \end{align}
Find the monthly charge at which both companies have the same fee.
- Let $c$ = the cost at which the charges from both companies are the same.
- Multiply the Number of Minutes (250) by the Unit Rate for Company B ($0.035).
\begin{align} c &= 250(0.035) &&\text{Given. Multiply the left side.} \tag 1 \\[2ex] c &= $8.75 &&\text{Solution.} \tag 2 \\[2ex] \end{align}
Solutions
- 250 minutes
- $8.75 per month