When writing a numeric expression we use numbers and basic operations to build up a number sentence that can be later calculated. Algebraic expressions are the same as numeric expressions except that they also use some new algebraic tools. These new algebraic tools are variables and coefficients.
We use algebraic expressions when we want to write a number sentence but don't know all the numbers involved.
For example: What is the total weight of a cat and a dog?
In this example we know that the total weight will be the weight of the cat added to the weight of the dog, but we don't know the number for either of these.
What we do instead is pretend that we know what these numbers are and replace them with variables.
In this case, let's use $c$c for the weight of the cat and $d$d for the weight of the dog.
Now we can write the number sentence as:
Total weight = $c+d$c+d
This is an algebraic expression as it is a number sentence that uses variables in the place of some numbers.
A variable is a symbol, commonly a letter, that is used in the place of a numeric value.
Let's try converting a word problem into an algebraic expression with the help of some variables.
There are some red fish and some blue fish in a tank. If $5$5 yellow fish are added to the tank, how many fish are now in the tank? Write an equation or expression for this scenario.
Think: We don't know how many red fish or blue fish there are in the tank so instead we can use variables to represent their numbers.
Do: Let's use $r$r for the red fish and $b$b for the blue fish. This means that:
Using these variables we can write an algebraic expression for the total number of fish in the tank:
Total number of fish = $r+b+5$r+b+5
Reflect: Since we didn't have numeric values for the number of red or blue fish we simply replaced them with variables. We then wrote the algebraic expression for the total number of fish as a sum using these variables. Notice that we couldn't evaluate $r+b$r+b as they are different variables with unknown values.
It should be noted that the choice of $r$r and $b$b as the variables is arbitrary and we can use whatever symbol we want for our variables, provided they don't already represent a value.
Coefficients are used in algebraic expressions to represent how many sets of a variable we have. They are written in front of a variable without a multiplication symbol like so:
Notice how we don't need the multiplication symbol to represent multiple sets of a variable.
This is because there is no danger of mixing up a coefficient next to a variable with any other term, whereas if we did this for numbers they would get mixed up with two digit numbers (for example: $3\times4=12\ne34$3×4=12≠34).
A coefficient is a numeral that is placed before and multiplies a variable in an algebraic term.
Coefficients are a bit different from multiplication though, since they also include the sign of the term.
This means that a negative term, $-6q$−6q for example, has a coefficient of $-6$−6.
We can see this more clearly in a longer expression.
Consider the expression: $4x-3y+7z$4x−3y+7z
By breaking up the expression into its individual terms we can determine the coefficients of each variable.
Term | Coefficient | Variable |
---|---|---|
$4x$4x | $4$4 | $x$x |
$-3y$−3y | $-3$−3 | $y$y |
$+7z$+7z | $7$7 | $z$z |
From this we can see that the coefficient of $y$y is $-3$−3, since it is a negative term, and the coefficient of $z$z is $7$7, since it is a positive term. If there is no sign in front of a term we assume that the term is positive, so we know that the coefficient of $x$x is $4$4.
We can also have algebraic terms where the coefficient is a fraction.
Consider: $v\div4$v÷4$=$=$\frac{v}{4}$v4$=$=$\frac{1}{4}\times v$14×v.
Since dividing by a number is the same as multiplying by its reciprocal, dividing by $4$4 gives us a coefficient of $\frac{1}{4}$14.
What about variables that don't appear to have coefficients?
Consider the term $x$x.
Since $x$x is equal to $1\times x$1×x which is also equal to $1x$1x, it actually has a coefficient of $1$1.
Whenever a variable has no written coefficient, its coefficient can be assumed to be $1$1.
Similarly, the coefficient of $-x$−x is $-1$−1.
Aside from the use of coefficients in multiplication, the basic operations work almost the same for algebraic terms as they do for numbers.
Between variables and numbers we have:
Word Expression | Algebraic Expression | Simplified Algebraic Expression | Represented with Algebra Tiles |
---|---|---|---|
three more than $x$x | $x+3$x+3 | $x+3$x+3 | |
three less than $x$x | $x-3$x−3 | $x-3$x−3 | |
the quotient of $x$x and three | $x\div3$x÷3 | $\frac{x}{3}$x3 | |
the product of $x$x and three | $x\times3$x×3 | $3x$3x |
One way to represent algebraic expressions is with algebra tiles. Negative values are often represented with the color red. Terms that can be combined are represented in the same size shape. Terms that cannot be combined are different sizes.
Positive Terms | Negative Terms |
---|---|
If $x$x represents the number of peaches then write an expression for the number of peaches minus $17$17.
If $x$x represents the number of pencils then write an expression for the number of pencils divided by $13$13.
Write expressions that record operations with numbers and with letters standing for numbers.
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.