What if you had a jar filled with dimes and quarters? You know that the total of the coins in the jar is $8.60. How could you write an equation to represent this situation? After completing this Concept, you'll be able to use variables to write equations like this one with unknown quantities.
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CK-12 Foundation: 0101S Language of Algebra
Guidance
No one likes doing the same problem over and over again—that’s why mathematicians invented algebra. Algebra takes the basic principles of math and makes them more general, so we can solve a problem once and then use that solution to solve a group of similar problems.
In arithmetic, you’ve dealt with numbers and their arithmetical operations (such as \begin{align*}+, \ -, \ \times, \ \div\end{align*} ). In algebra, we use symbols called variables (which are usually letters, such as @$\begin{align*}x, \ y, \ a, \ b, \ c, \ \ldots\end{align*}@$ ) to represent numbers and sometimes processes.
For example, we might use the letter @$\begin{align*}x\end{align*}@$ to represent some number we don’t know yet, which we might need to figure out in the course of a problem. Or we might use two letters, like @$\begin{align*}x\end{align*}@$ and @$\begin{align*}y\end{align*}@$ , to show a relationship between two numbers without needing to know what the actual numbers are. The same letters can represent a wide range of possible numbers, and the same letter may represent completely different numbers when used in two different problems.
Using variables offers advantages over solving each problem “from scratch.” With variables, we can:
- Formulate arithmetical laws such as @$\begin{align*}a + b = b + a \end{align*}@$ for all real numbers @$\begin{align*}a\end{align*}@$ and @$\begin{align*}b\end{align*}@$ .
- Refer to “unknown” numbers. For instance: find a number @$\begin{align*}x\end{align*}@$ such that @$\begin{align*}3x + 1 = 10\end{align*}@$ .
- Write more compactly about functional relationships such as, “If you sell @$\begin{align*}x\end{align*}@$ tickets, then your profit will be @$\begin{align*}3x - 10\end{align*}@$ dollars, or “ @$\begin{align*}f(x) = 3x - 10\end{align*}@$ ,” where “ @$\begin{align*}f\end{align*}@$ ” is the profit function, and @$\begin{align*}x\end{align*}@$ is the input (i.e. how many tickets you sell).
Example A
Write an algebraic equation for the perimeter and area of the rectangle below.
To find the perimeter, we add the lengths of all 4 sides. We can still do this even if we don’t know the side lengths in numbers, because we can use variables like @$\begin{align*}l\end{align*}@$ and @$\begin{align*}w\end{align*}@$ to represent the unknown length and width. If we start at the top left and work clockwise, and if we use the letter @$\begin{align*}P\end{align*}@$ to represent the perimeter, then we can say:
@$$\begin{align*} P = l + w + l + w\end{align*}@$$
We are adding @$\begin{align*}2 \ l\end{align*}@$ ’s and @$\begin{align*}2 \ w\end{align*}@$ 's, so we can say that:
@$$\begin{align*}P = 2 \cdot l + 2 \cdot w\end{align*}@$$
It's customary in algebra to omit multiplication symbols whenever possible. For example, @$\begin{align*}11x\end{align*}@$ means the same thing as @$\begin{align*}11 \cdot x\end{align*}@$ or @$\begin{align*}11 \times x\end{align*}@$ . We can therefore also write:
@$$\begin{align*} P = 2l + 2w\end{align*}@$$
Area is length multiplied by width . In algebraic terms we get:
@$$\begin{align*} A = l \times w \to A = l \cdot w \to A = lw\end{align*}@$$
Note: @$\begin{align*}2l + 2w\end{align*}@$ by itself is an example of a variable expression ; @$\begin{align*} P = 2l + 2w \end{align*}@$ is an example of an equation . The main difference between expressions and equa tions is the presence of an equa ls sign (=).
In the above example, we found the simplest possible ways to express the perimeter and area of a rectangle when we don’t yet know what its length and width actually are. Now, when we encounter a rectangle whose dimensions we do know, we can simply substitute (or plug in ) those values in the above equations. In this chapter, we will encounter many expressions that we can evaluate by plugging in values for the variables involved.
Example B
Eric has some money in his savings account. How much more money does he need in order to buy a game that costs $98?
Solution:
Let @$\begin{align*}M\end{align*}@$ be the money that Eric still needs and let @$\begin{align*}S\end{align*}@$ be the money that Eric has in his savings account. Then, by subtracting the money he already has from the total money needed, we can figure out how much money he still needs:
@$\begin{align*}M=98-S.\end{align*}@$
Example C
Write an equation for the sum of 3 times some number and 5.
Solution:
Let @$\begin{align*}S\end{align*}@$ be the total sum. Let @$\begin{align*}n\end{align*}@$ be some number. Then 3 times some number is @$\begin{align*} 3\cdot n\end{align*}@$ and then the sum of that and 5 is:
@$\begin{align*}S=3n+5. \end{align*}@$
Watch this video for help with the Examples above.
CK-12 Foundation: The Language of Algebra
Vocabulary
- variables : Variables are symbols (which are usually letters, such as @$\begin{align*}x, \ y, \ a, \ b, \ c, \ \ldots\end{align*}@$ ) used to represent numbers and sometimes processes.
- variable expression : A variable expression is a group of variables, constants (numbers), and operations that does not include an equals (=) sign. @$\begin{align*}2l + 2w\end{align*}@$ is an example of an expression.
- equation : An equation is a group of variables, constants (numbers), and operations that does include an equals (=) sign. @$\begin{align*} P = 2l + 2w \end{align*}@$ is an example of an equation.
Guided Practice
Alex has a certain amount of nickels and dimes in a jar. Write an algebraic equation for how much money she has, in terms of how many nickles and dimes she has.
Solution:
Let @$\begin{align*}n\end{align*}@$ be the number of nickels and @$\begin{align*}d\end{align*}@$ be the number of dimes that Alex has in the jar. Since each nickel is worth $0.05, the amount of money she has in nickels will be:
@$\begin{align*}0.05\cdot n\end{align*}@$
Since each dime is worth $0.10, the amount of money she has in dimes will be:
@$\begin{align*}0.10\cdot d\end{align*}@$
This means that the total amount of money @$\begin{align*}M\end{align*}@$ that Alex has will be:
@$\begin{align*}M=0.05 \cdot n +0.10 \cdot d.\end{align*}@$
Simplifying the expressions, we get:
@$\begin{align*}M=0.05n +0.10d.\end{align*}@$
Explore More
For 1-4, write the following in a more condensed form by leaving out a multiplication symbol.
- @$\begin{align*}2 \times 11x\end{align*}@$
- @$\begin{align*}1.35 \cdot y\end{align*}@$
- @$\begin{align*}3 \times \frac { 1 } { 4 } \end{align*}@$
- @$\begin{align*}\frac { 1 } { 4 } \cdot z \end{align*}@$
For 5-10, write an equation for the following situations.
- The amount of money Andrea has in a jar full of quarters and dimes.
- The amount of money Michelle has in her coin purse if it only contains quarters, dimes and pennies.
- The sum of 7 and 6 times some number.
- 4 less than 20 times some number.
- The amount of money you will earn if you are paid $10.25 an hour and spend $4.00 round trip to get to and from work.
- A father earns a $2000 dividend from an oil investment and distributes it equally amongst his children.