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.
CK-12 Foundation: 0101S Language of Algebra
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.
Using variables offers advantages over solving each problem “from scratch.” With variables, we can:
- Formulate arithmetical laws such as a+b=b+a for all real numbers a and b.
- Refer to “unknown” numbers. For instance: find a number x such that 3x+1=10.
- Write more compactly about functional relationships such as, “If you sell x tickets, then your profit will be 3x−10 dollars, or “f(x)=3x−10,” where “f” is the profit function, and x is the input (i.e. how many tickets you sell).
Write an algebraic equation for the perimeter and area of the rectangle below.
Area is length multiplied by width. In algebraic terms we get:
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.
Eric has some money in his savings account. How much more money does he need in order to buy a game that costs $98?
Write an equation for the sum of 3 times some number and 5.
Watch this video for help with the Examples above.
CK-12 Foundation: The Language of Algebra
- We use symbols called variables (which are usually letters, such as x, y, a, b, c, …) to represent numbers and sometimes processes.
2l+2w by itself is an example of a variable expression; P=2l+2w is an example of an equation. The main difference between expressions and equations is the presence of an equals sign (=).
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.
Since each dime is worth $0.10, the amount of money she has in dimes will be:
Simplifying the expressions, we get:
For 1-4, write the following in a more condensed form by leaving out a multiplication symbol.
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 too and from work.
- A father earns a $2000 dividend from an oil investment and distributes it equally amongst his children.