Problem 43 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each expression without parentheses. Then simplify the result. See Example 4. \(-7(3 y+5)\)

Step-by-Step Solution

Answer

The expression simplifies to \(-21y - 35\).

Step 1: Apply Distributive Property

The distributive property states that for all real numbers, \(a(b + c) = ab + ac\). Using this property on \(-7(3y + 5)\), we distribute \(-7\) to both terms inside the parentheses: \(-7 \times 3y\) and \(-7 \times 5\).

Step 2: Calculate Each Product

After distributing, calculate each product separately. For \(-7 \times 3y\), multiply \(-7\) by \(3y\) to get \(-21y\). For \(-7 \times 5\), multiply \(-7\) by \(5\) to get \(-35\).

Step 3: Write and Simplify the Expression

Write the expression by combining the results from Step 2: \(-21y - 35\). Since there are no like terms to combine in this expression, it is already simplified.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their most basic form. This often means performing operations like multiplication, addition, or subtraction to remove any unnecessary parts of the expression.
In the example exercise, this is done by applying the distributive property. Once the expression \(-7 (3y + 5)\) is distributed, we get \(-21y - 35\). There are no like terms to combine further, so it can’t be simplified further.
Here are a few steps to follow when simplifying expressions:

• Distribute any numbers outside the parentheses.
• Combine like terms, if possible.
• Simplify any fractions or complex terms.

Following these simple steps can make any complex algebraic expression easier to understand. Keep practicing to become more confident in these simplifications!

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, or division. They are fundamental in mathematics and act as instructions for combining numbers and variables.
In the case of the given exercise, \(-7(3y + 5)\), we work with a mix of numbers and the variable \(y\). The purpose is to perform mathematical operations to simplify it.
Algebraic expressions can be basic with just one term (like \(5x\)) or more complex with many terms and operations (like \(3x^2 + 4x - 7\)). Here are a few common features:

• Terms: These are the parts of an expression separated by + or - signs, for example, \(3x + 2\) has two terms.
• Coefficients: Numbers in front of variables - in \(3x\), 3 is the coefficient.
• Constants: Numbers on their own, like 5 in \(3x + 5\).

Understanding the structure and parts of algebraic expressions helps in simplifying and solving them.

Real Numbers

Real numbers include all the numbers you might come across on the number line. They include rational numbers, like integers (whole numbers) and fractions, as well as irrational numbers, like pi or the square root of 2.
In the algebraic world, real numbers are the coefficients of terms, as seen in the expression \(-7(3y + 5)\). Here, -7, 3, and 5 are real numbers. Real numbers are crucial because they allow us to apply operations like addition, subtraction, and multiplication to solve expressions. Some key properties include:

• They have an ordering, which means you can compare any two real numbers to find which is greater or lesser.
• They are dense, meaning between any two real numbers, there's always another real number.
• They are used to practically measure real-world quantities like distance and temperature.

Grasping real numbers' properties helps understand how they interact with variables, such as \(y\), in algebraic expressions. The ability to work with these numbers is foundational in solving algebraic problems efficiently.