Lesson

Introduction to Expressions and Variables

Welcome to the world of Algebra! One of the first things you'll encounter are expressions and variables. These are the building blocks for more complex algebraic concepts. In this lesson, we'll break down what they are and how to work with them.

What is a Variable?

A variable is a symbol, usually a letter, that represents an unknown number. Think of it as a placeholder. We use variables when we don't know the exact value of something, or when the value can change. Common variables are \( x \) , \( y \) , \( z \) , \( a \) , \( b \) , and so on.

For example, in the statement "a number plus 5 equals 10", we could represent "a number" with the variable \( x \) . So, the statement becomes \( x+5=10 \) .

What is an Expression?

An expression is a combination of numbers, variables, and mathematical operations (like addition, subtraction, multiplication, and division). It doesn't have an equals sign (=). It's just a way to represent a mathematical idea.

Some examples of expressions are: \( 3+x \) , \( 5y-2 \) , \( a\times b \) , and \( \frac{z}{4} \) .

Building Simple Expressions

Let's practice building some simple expressions from word phrases:

1. "A number increased by 7": We can represent this as \( x+7 \) .
2. "Twice a number": This means 2 multiplied by the number, so we can write it as \( 2y \) .
3. "A number divided by 3": This would be written as \( \frac{z}{3} \) or \( z\div 3 \) .
4. "Five less than a number": This is \( a-5 \) . (Important: The order matters here!)

More Complex Expressions

Expressions can involve more than one operation. For example:

1. "Three times a number, plus 4": This is \( 3x+4 \) .
2. "Half of a number, minus 1": This is \( \frac{y}{2}-1 \) or \( \frac{1}{2}y-1 \) .
3. "The sum of two numbers": If the two numbers are \( a \) and \( b \) , this is \( a+b \) .

Understanding Coefficients

A coefficient is the number that is multiplied by a variable. In the expression \( 5y-2 \) , 5 is the coefficient of the variable \( y \) . If a variable appears alone, like \( x \) , its coefficient is understood to be 1 (because \( 1\times x=x \) ).

Constants

A constant is a number that doesn't change its value. In the expression \( 5y-2 \) , -2 is a constant. Constants are terms in an expression that don't have any variables attached to them.

Terms

Terms are the parts of an expression that are separated by addition or subtraction. In the expression \( 3x+4 \) , \( 3x \) and \( 4 \) are the terms. In the expression \( 5y-2 \) , \( 5y \) and \( -2 \) are the terms.

Putting it all Together: Examples

Let's look at some more examples and identify the different parts:

Evaluating Expressions

Evaluating an expression means finding its value when you know the values of the variables. To do this, you simply substitute the given values for the variables and perform the indicated operations.

Example of Evaluating Expressions

Let's say we have the expression \( 3x+2 \) , and we know that \( x=4 \) . To evaluate the expression, we substitute 4 for \( x \) :

So, the value of the expression \( 3x+2 \) when \( x=4 \) is 14.

Another Evaluation Example

Consider the expression \( y^2-5 \) , where \( y=3 \) . (Note: \( y^2 \) means \( y \) squared, or \( y\times y \) ). Substitute 3 for \( y \) :

Therefore, the value of the expression \( y^2-5 \) when \( y=3 \) is 4.

Practice Makes Perfect

The best way to get comfortable with expressions and variables is to practice. Try creating your own expressions from word phrases and evaluating them with different values for the variables. The more you practice, the easier it will become!

Summary

In this lesson, we've covered the basics of expressions and variables:

• A variable is a symbol (usually a letter) that represents an unknown number.
• An expression is a combination of numbers, variables, and mathematical operations, without an equals sign.
• Coefficients are the numbers multiplied by variables.
• Constants are numbers that don't change their value.
• Terms are the parts of an expression separated by addition or subtraction.
• Evaluating an expression means finding its value by substituting given values for the variables.

Next Steps

Now that you have a good grasp of expressions and variables, you're ready to move on to more advanced topics in Algebra, such as simplifying expressions and solving equations. Keep practicing and have fun!