65 Unit Expressions And Equations Homework 2 Answer Key

1. Simplifying Expressions

1.1 Combining Like Terms

When simplifying expressions, it is important to combine like terms. Like terms are terms that have the same variable raised to the same exponent. To combine like terms, add or subtract the coefficients while keeping the variable and exponent the same. For example, in the expression 3x + 2x - 5x, the like terms are 3x, 2x, and -5x. Combining them gives us 3x + 2x - 5x = 0x, which simplifies to 0.

1.2 Distributive Property

The distributive property is another important concept when simplifying expressions. It states that when you multiply a number by a sum or difference, you can distribute the multiplication to each term inside the parentheses. For example, in the expression 2(3x + 4), we can distribute the 2 to both terms inside the parentheses, resulting in 6x + 8.

2. Solving Equations

2.1 One-Step Equations

One-step equations are equations that can be solved in just one step. To solve a one-step equation, isolate the variable by performing the inverse operation. For example, in the equation 3x = 12, we can solve for x by dividing both sides by 3. This gives us x = 4.

2.2 Two-Step Equations

Two-step equations require two operations to solve. Begin by performing the inverse of addition or subtraction, and then apply the inverse of multiplication or division. For example, in the equation 2x + 5 = 13, we can first subtract 5 from both sides to get 2x = 8. Then, divide both sides by 2 to find x = 4.

3. Applying Algebraic Concepts

3.1 Word Problems

Algebraic concepts can be applied to solve word problems. These problems often involve setting up equations and solving for unknown variables. For example, if the problem states "The sum of two numbers is 15, and one number is 3 more than the other," we can represent this with the equation x + (x + 3) = 15, where x represents the smaller number. Solving this equation gives us x = 6, so the smaller number is 6 and the larger number is 9.

3.2 Order of Operations

When working with expressions, it is important to follow the order of operations. The acronym PEMDAS can help you remember the correct order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). For example, in the expression 3 + 4 * 2, we would first perform the multiplication, giving us 3 + 8 and then the addition, resulting in 11.

4.1 Question 1

Explanation: The expression 2x + 3 - x can be simplified by combining like terms. The x terms cancel out, leaving us with 3.

4.2 Question 2

Explanation: To solve the equation 2x + 1 = 15, we can subtract 1 from both sides to get 2x = 14. Then, divide both sides by 2 to find x = 7.

4.3 Question 3

Explanation: In the expression 3(2x + 4), we can distribute the 3 to both terms inside the parentheses, resulting in 6x + 12. Simplifying further, we get 6x + 12 = 6x - 2x + 22. Combining like terms gives us 4x + 12 = 22. Subtracting 12 from both sides gives us 4x = 10, and dividing by 4 gives us x = 2.5.