Problem 7
Question
Use the distributive property to rewrite the expression without parentheses. $$ (12-x) y $$
Step-by-Step Solution
Verified Answer
The expression (12-x) y simplifies to \: 12y - xy
1Step 1: Distribute \(y\) to \(12\)
First, distribute the term outside the parentheses \(y\) by multiplying it with the first term inside the parentheses which is \(12\). This gives the term \(12y\).
2Step 2: Distribute \(y\) to \(-x\)
Next, multiply again the term outside the parentheses \(y\) with the second term inside the parentheses which is \(-x\). This gives the term \(-xy\).
3Step 3: Write down the simplified expression
Now that we have multiplied \(y\) with each term inside the parentheses correctly, we just have to write down the result which is the sum of the terms we found: \(12y - xy\)
Key Concepts
Simplifying ExpressionsAlgebraic ExpressionsMultiplying Variables
Simplifying Expressions
In algebra, simplifying expressions is a fundamental process where we alter the form of an expression without changing its value. Simplifying makes equations and other algebraic tasks easier to perform and understand. When you come across a problem like \( (12-x) y \), simplifying involves distributing the \( y \) across the terms inside the parentheses. You multiply \( y \) with each term, which essentially spreads out the multiplication.
Imagine you have a bag of apples represented by \( y \) and two boxes with 12 and \( -x \) apples respectively. Distributing \( y \) is like deciding to give \( y \) apples to each box, which gives you \(12y\) apples from the first and \( -xy \) from the second. The combined amount of apples from both boxes is what you have after simplification: \( 12y - xy \). It's just organizing your apples in a clear way!
Imagine you have a bag of apples represented by \( y \) and two boxes with 12 and \( -x \) apples respectively. Distributing \( y \) is like deciding to give \( y \) apples to each box, which gives you \(12y\) apples from the first and \( -xy \) from the second. The combined amount of apples from both boxes is what you have after simplification: \( 12y - xy \). It's just organizing your apples in a clear way!
Algebraic Expressions
An algebraic expression combines numbers, variables, and arithmetic operations (like addition, subtraction, multiplication, and division) according to the rules of arithmetic and algebra. Variables are symbols (often letters) that represent numbers, and they allow for generalizations in math.
Think about a variable like a placeholder for any number, somewhat like a wildcard. The expression \( 12y - xy \) has two terms: \(12y\) and \(xy\). Here, \( y \) and \( x \) are variables. This type of expression can model countless scenarios, like calculating costs, mixing ingredients, or predicting outcomes. By understanding how the variables relate to one another, we can describe and solve real-world problems with precision.
Think about a variable like a placeholder for any number, somewhat like a wildcard. The expression \( 12y - xy \) has two terms: \(12y\) and \(xy\). Here, \( y \) and \( x \) are variables. This type of expression can model countless scenarios, like calculating costs, mixing ingredients, or predicting outcomes. By understanding how the variables relate to one another, we can describe and solve real-world problems with precision.
Multiplying Variables
The concept of multipliyng variables follows the same basic principles of multiplying numbers. When you multiply variables, you are combining their quantities in a way to express a new quantity. If you have a variable \( a \) representing a certain number of items and another variable \( b \) representing another set, \( ab \) means you're taking the quantity of \( a \) items, \( b \) times.
For example, when we say \( -xy \) from the earlier expression \( (12-x) y \), it's like taking the quantity \( x \) and expanding it by \( y \) times, but with the twist that \( x \) is subtracted (as indicated by the minus sign), showing that multiplication isn't just about 'increasing' but could represent 'combining' in various ways—like mixtures, areas of shapes, or changing directions.
For example, when we say \( -xy \) from the earlier expression \( (12-x) y \), it's like taking the quantity \( x \) and expanding it by \( y \) times, but with the twist that \( x \) is subtracted (as indicated by the minus sign), showing that multiplication isn't just about 'increasing' but could represent 'combining' in various ways—like mixtures, areas of shapes, or changing directions.