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 case, let's use c for the weight of the cat and d for the weight of the dog.
c + d is called an algebraic expression.
c and d are called variables. They are used in the place of a numeric value.
Coefficients are used to show how many variables we have. The variable u with a coefficient of 3 is written as 3u which means 3 \cdot u.
Terms are the building blocks of an expression. They are either a single number or variable or numbers and variables multiplied together. Terms are separated by + or - signs.
Consider the expression: {4x-3y} + \frac{2z}{5} + k + 7
This is an expression with 5 terms.
The term 4x has a coefficient of 4.
The term -3y has a coefficient of -3. The negative belongs with the coefficient.
The term \dfrac{2z}{5} has a coefficient of \dfrac{2}{5} as \dfrac{2z}{5} = \dfrac{2}{5} \cdot z.
k is equal to 1 \cdot k which is also equal to 1k, so it has a coefficient of 1.
The term 7 has no variable. It is called a constant term.
In order to write an expression that can be used to model the total cost of buying new school supplies, Mr. Okware defines the variables:
Let x represent the cost of a folder, y represent the cost of a calculator, and z represent the cost of a pencil pack.
What could these expressions represent in this context?
In this context, what do the coefficients describe?
Expressions and parts of expressions, like factors and coefficients, all have unique meanings in a given context. Viewing expressions in parts and as a whole while paying attention to the quantities represented by the variables can explain the relationships described by the expressions.
For the algebraic expression 5x+7y-12x+21:
Determine the number of terms.
Identify the coefficient of the first term.
Identify the constant term.
Determine if the expression contains like terms.
A courier service charges a fixed fee of \$30 plus \$0.75 per kilogram for delivering packages. Write an algebraic expression for the total delivery cost of a package weighing m kilograms.
Write an algebraic expression for the phrase "eight more than the quotient of 9 and x".
The perimeter of a rectangle can be expressed by 2 \left(l+w \right). Explain what each of the factors represents.
Expressions can be used to represent mathematical relationships. In an expression, sums often represent totals and coefficients and factors represent multiplication. When interpreting an expression in context, we can use the units to help understand the meaning.
We can use algebra tiles to help us visualize algebraic expressions.
The tile x represents an unknown number. The tile +1 represents adding one unit and -1 represents subtracting one unit.
This table demonstrates how expressions can be built using the tiles:
We can use visual models to rearrange expressions into equivalent expressions. We can see how each of the following equivalent expressions is represented with the algebra tiles.
All of these expressions are equivalent to 6x+6. Different grouping variations of equivalent expressions can help us understand them better in different mathematical situations.
Write an equivalent algebraic expression for the following:
Represent the expression -3x-5+8 using algebra tiles.
We can represent algebraic expressions with visual models to better understand them.
We can rearrange models of algebraic expressions to generate equivalent expressions.