In this unit, students will explore how to solve linear equations by combining like terms, using the distributive property, and identifying the number of solutions for the equations.
Term: A number, a variable(s), or a number and a variable(s) multiplied together. Terms are separated by a positive (+) or negative (-) sign.
Constant: A number by itself with no variable.
Coefficient: The number being multiplied by the variable.
Like Terms: Terms that have identical variable(s).
Unlike Terms: Terms that do not have identical variable(s).
Click on each word to see an example: Term, Constant, Coefficient, Like Terms, Unlike Terms.
In algebra, terms are the basic components of expressions. A term can be a single number (a constant), a variable (like x or y), or a combination of numbers and variables multiplied together. Terms are separated by addition (+) or subtraction (-) signs in an expression. Understanding the distinction between like and unlike terms is crucial as it helps in simplifying expressions and solving algebraic equations effectively.
Like terms are terms that have exactly the same variable components raised to the same powers. The coefficients (the numbers multiplying the variables) do not need to be the same. For instance, in the expression 3x2 + 5x2, both terms are like terms because they contain the same variable x raised to the same power (2).
Examples of like terms include:
Unlike terms, on the other hand, are terms that have different variable parts or exponents. The difference might be in the variable itself or in the exponent of the same variable. For example, 3x2 and 5x3 are unlike terms because, even though they both contain the variable x, the powers are different.
More examples include:
Simplifying algebraic expressions often involves combining like terms. This process is called "collecting like terms." For example, in the expression 4x + 3x2 - 2x, you can combine the like terms 4x and -2x to simplify the expression to 2x + 3x2.
When dealing with unlike terms, remember that they cannot be combined through addition or subtraction due to their different variable parts or exponents. Each term stands alone in the expression.
| Expression | Like or Unlike? |
|---|---|
| 3x and 5x | Like |
| 2x² and 3x | Unlike |
| 7y and -7y | Like |
| 4ab and 4ba | Like |
| 5m and 5n | Unlike |
Practice entering 'like' or 'unlike' for the following expressions:
3x and 2x
5x and 5y
6a and 6a
4bc and 4cb
7x² and 7x
Combining like terms is a critical process in algebra that involves simplifying algebraic expressions by adding or subtracting terms that are alike. Terms are considered 'like' if they have identical variable parts, including the variables and their exponents.
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The primary reason for combining like terms is to simplify an expression, making it easier to handle and solve equations. Simplification can help in solving equations, evaluating expressions, and understanding algebraic relationships more clearly.
Like terms have the same variable factors. For example, 2x2y and 5x2y are like terms because both contain the same variables x and y, with x raised to the 2nd power. However, 2xy and 2x2y are not like terms because the exponent of x differs.
Consider the expression 3x + 4x - 2x + 5. Here, all the terms involving x are like terms. They can be combined as follows:
The simplified expression is 5x + 5.
Let's simplify the expression 6a2b - 3ab2 + 2a2b + 4ab2:
The simplified expression is 8a2b + ab2.
Combining like terms is fundamental in algebra for simplifying expressions and solving equations more efficiently. It helps to create clearer and less complex expressions which are crucial for advanced algebraic operations and calculus.
| Expression | Simplified |
|---|---|
| 3x + 2x | 5x |
| 4a + 6a | 10a |
| 7y - 2y | 5y |
| 5b + 3b + 2b | 10b |
| 9m - 4m | 5m |
Practice simplifying the following expressions:
5x + 3x
6y + 2y - 3y
10z - 6z
12a + 7a + 1a
9m - 5m + 4m
The Distributive Property, also known as the distributive law of multiplication over addition, is a crucial algebraic property used to simplify expressions and solve equations. It states that multiplying a sum by a number is the same as multiplying each addend of the sum by the number and then adding the results.
The formula for the distributive property can be expressed as follows:
a(b + c) = ab + ac
This property is particularly useful for simplifying expressions that involve parentheses and are not immediately easy to add or subtract. It helps in breaking down complex problems into simpler parts that are easier to manage and solve.
Consider the expression 3(2 + 4). Applying the distributive property, we multiply 3 by each addend inside the parentheses:
This simplifies the expression to 18, demonstrating how the distributive property aids in multiplication across a sum.
The distributive property is also invaluable when dealing with variable expressions. For example:
Let’s apply the distributive property to a more complex expression: 4(x + 2y - 3z). Multiplying 4 by each term within the parentheses gives:
This demonstrates how the property simplifies expressions involving multiple terms and different operations.
The distributive property is essential for simplifying and solving algebraic equations, making it a foundational element in algebra and beyond. Understanding and applying this property allows for more efficient and accurate problem-solving in mathematics.
| Expression | Simplified |
|---|---|
| 2(3x + 4) | 6x + 8 |
| 5(2y + 3) | 10y + 15 |
| 3(4z - 2) | 12z - 6 |
| 4(1a + 2b) | 4a + 8b |
| 6(1m - 3n) | 6m - 18n |
Practice applying the distributive property to these expressions:
3(2x + 4)
4(3y + 5)
5(2z - 3)
2(4a + 6b)
7(3m + 2n)