Question

Identify an unknown and rewrite each expression as an algebraic expression. Ten more than three times a number.

Short answer

The algebraic expression is: \( 3x + 10 \).

Step-by-step solution

Identify the unknown

Choose a variable to represent the unknown number. Often, we use the letter 'x' to represent an unknown variable in algebra.

Translate into algebraic terms

Translate the words 'ten more than three times a number' into an algebraic expression. 'Three times a number' suggests multiplication (3 * x), and 'ten more than' suggests addition (+ 10).

Key concepts

Algebraic Expression

An algebraic expression is a combination of letters (variables), numbers and at least one arithmetic operation (like addition, subtraction, multiplication, or division). In our example, 'ten more than three times a number,' we are constructing an algebraic expression that captures this relationship.

To translate this, we start with 'three times a number,' which becomes the multiplication part of our expression, represented as '3 times x' or '3x'. Then, we add the 'ten more than,' making it '3x + 10'. This entire entity is our algebraic expression. It's important because it lets us work with numerical relationships before we know all the specific values.

Unknown Variable

In algebra, an unknown variable is a symbol that stands in for a number we don't yet know. It's like a placeholder. By tradition, we often use letters towards the end of the alphabet, like 'x', 'y', or 'z', to name these unknowns. For example, if we are talking about 'a number' in a problem, we might let 'x' be that number. When we say 'ten more than three times a number', we don't know what 'a number' is, so we call it 'x'. Later on, we might figure out what 'x' actually represents by solving the algebraic expression.

Algebraic Terms

When we break down an algebraic expression, we find it consists of algebraic terms. Each term is either a single number (a constant), a single variable, or a combination of numbers and variables that are multiplied or divided.

In our example, '3x' is one term and '10' is another. The term '3x' means 3 multiplied by the variable 'x', and '10' is just a number by itself. Combining these, we get '3x + 10', which means 'three times some number plus ten.' Collectively, terms are the building blocks of algebraic expressions and understanding each part helps decipher the whole.

Basic Algebra

Finally, basic algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's a tool for representing and solving problems involving unknown numbers. Using what we've learned, we take real-world scenarios and construct equations or expressions that help us solve for unknowns.

For instance, if 'three times a number plus ten equals forty', in basic algebra, we'd write \(3x + 10 = 40\). To find the unknown 'x', we'd use algebraic principles like inverse operations to isolate 'x' and discover its value. The ultimate goal of basic algebra is to understand and solve for these unknowns, making it a fundamental skill in math.