Problem 42 - Prealgebra

Question

Apply the distributive property to expression, and then simplify. \(7(5 x-y)\)

Step-by-Step Solution

Answer

The simplified expression is \(35x - 7y\).

Step 1: Apply the Distributive Property

The distributive property states that for any numbers or expressions "a", "b", and "c", the expression \(a(b+c)\) is equal to \(ab + ac\). We apply this property to the given expression \(7(5x - y)\). Here, "7" is "a", "5x" is "b", and "-y" is "c". This gives us:\[7(5x - y) = 7 imes 5x + 7 imes (-y)\]

Step 2: Multiply the Terms

Now, perform the multiplication for each term:1. Multiply 7 by \(5x\): \(7 imes 5x = 35x\)2. Multiply 7 by \(-y\): \(7 imes (-y) = -7y\)So, after distribution, we have:\[7(5x - y) = 35x - 7y\]

Step 3: Simplify the Expression

Since there are no like terms to combine in \(35x - 7y\), the expression is already simplified. The final simplified version of the expression is:\[35x - 7y\]

Key Concepts

Simplifying Algebraic Expressions

When simplifying algebraic expressions, our main goal is to express the given terms in the simplest form. This often involves combining like terms and using the distributive property. In our example, we were dealing with the expression \(7(5x - y)\). To simplify it, we followed these important steps:

• First, we used the distributive property to "distribute" the 7 to both terms inside the parentheses (5x and -y). This is because the distributive property allows us to multiply each term individually by the term outside the parentheses.
• Next, we performed the multiplications: \(7 \times 5x\) and \(7 \times (-y)\). By solving these multiplications, we obtained \(35x\) and \(-7y\) respectively.
• Finally, since \(35x\) and \(-7y\) are not like terms, they cannot be further combined or simplified, resulting in the final expression \(35x - 7y\).

To master simplifying algebraic expressions, practice identifying like terms and apply distributive property rules effectively.

Prealgebra

Prealgebra serves as a fundamental building block for higher-level mathematics. It introduces basic algebraic concepts such as variables, operational symbols, and equations. Understanding these concepts is crucial before moving on to more advanced algebra.
In prealgebra, we often encounter expressions and need to simplify them to make them easier to understand or solve. For instance, using the distributive property - a key concept in prealgebra - we can simplify complex expressions by breaking them into manageable parts.

• A variable, like \(x\) in \(5x\), represents an unknown value and can take any number.
• Understanding operations like addition, subtraction, and multiplication in conjunction with variables prepares you for algebra.
• Recognizing patterns and practicing with different problems helps solidify these foundational skills.

Prealgebra ensures you have the necessary skills to tackle more complex algebraic expressions and equations confidently.

Algebraic Multiplication

Algebraic multiplication involves the multiplication of terms involving variables, numbers, or both. In our example, \(7(5x - y)\), multiplication involves not just numbers but also variables. Here’s a simple way to understand it:

• The first step is to apply the distributive property, which helps break down expressions. This involved multiplying the constant 7 by each term within the parentheses, resulting in \(35x\) and \(-7y\).
• By multiplying a constant with a term containing a variable, like in \(7 \times 5x = 35x\), you learn how the constant affects the magnitude of the variable term.
• Despite involving different terms, the multiplication process remains consistent in structure: multiply the numbers, then affix the variable.

Understanding algebraic multiplication is essential for solving equations and simplifying expressions, and it forms the basis of many algebra problems. The more you practice, the easier it becomes to handle complex multiplicative expressions.