The image you provided illustrates the
Distributive Property of multiplication over addition and subtraction. The distributive property is a fundamental concept in algebra that allows us to simplify expressions by distributing a factor across terms inside parentheses. Let's break it down step by step:
Key Formulas Shown:
1.
Distributive Property for Addition:
\[
a(b + c) = ab + ac
\]
This means that multiplying \(a\) by the sum of \(b\) and \(c\) is the same as multiplying \(a\) by \(b\) and \(a\) by \(c\) separately, then adding the results.
2.
Distributive Property for Subtraction:
\[
a(b - c) = ab - ac
\]
This means that multiplying \(a\) by the difference of \(b\) and \(c\) is the same as multiplying \(a\) by \(b\) and \(a\) by \(c\) separately, then subtracting the results.
3.
Distributive Property with a Negative Factor (Addition):
\[
-a(b + c) = -ab - ac
\]
Here, the negative sign is distributed to both \(b\) and \(c\), resulting in the negation of each term.
4.
Distributive Property with a Negative Factor (Subtraction):
\[
-a(b - c) = -ab + ac
\]
In this case, the negative sign is distributed to both \(b\) and \(-c\). Distributing the negative sign to \(-c\) results in \(+ac\).
Explanation of Each Case:
#### 1. \(a(b + c) = ab + ac\)
-
Example: Suppose \(a = 2\), \(b = 3\), and \(c = 4\).
- Left-hand side: \(a(b + c) = 2(3 + 4) = 2 \cdot 7 = 14\).
- Right-hand side: \(ab + ac = 2 \cdot 3 + 2 \cdot 4 = 6 + 8 = 14\).
- Both sides are equal, confirming the property.
#### 2. \(a(b - c) = ab - ac\)
-
Example: Suppose \(a = 2\), \(b = 5\), and \(c = 3\).
- Left-hand side: \(a(b - c) = 2(5 - 3) = 2 \cdot 2 = 4\).
- Right-hand side: \(ab - ac = 2 \cdot 5 - 2 \cdot 3 = 10 - 6 = 4\).
- Both sides are equal, confirming the property.
#### 3. \(-a(b + c) = -ab - ac\)
-
Example: Suppose \(a = 2\), \(b = 3\), and \(c = 4\).
- Left-hand side: \(-a(b + c) = -2(3 + 4) = -2 \cdot 7 = -14\).
- Right-hand side: \(-ab - ac = -2 \cdot 3 - 2 \cdot 4 = -6 - 8 = -14\).
- Both sides are equal, confirming the property.
#### 4. \(-a(b - c) = -ab + ac\)
-
Example: Suppose \(a = 2\), \(b = 5\), and \(c = 3\).
- Left-hand side: \(-a(b - c) = -2(5 - 3) = -2 \cdot 2 = -4\).
- Right-hand side: \(-ab + ac = -2 \cdot 5 + 2 \cdot 3 = -10 + 6 = -4\).
- Both sides are equal, confirming the property.
Final Answer:
The distributive property formulas shown in the image are correct and can be summarized as follows:
\[
\boxed{
\begin{aligned}
1. & \quad a(b + c) = ab + ac \\
2. & \quad a(b - c) = ab - ac \\
3. & \quad -a(b + c) = -ab - ac \\
4. & \quad -a(b - c) = -ab + ac
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of distributive property of subtraction.
