Problem 6 - Algebra 1: Concepts and Skills

Question

Decide whether the equation is true or false. Use the distributive property to explain your answer. $$ (2+5) 3=2(3)+5(3) $$

Step-by-Step Solution

Answer

Yes, the given equation \( (2 + 5) * 3 = 2 * 3 + 5 * 3 \) is true. This is justified by the use of the distributive property.

Step 1: Apply Distributive Property

According to the distributive property, \(a * (b + c) = a * b + a * c\). If we look at the given equation \((2 + 5) * 3 = 2 * 3 + 5 * 3\), we can see that in the left side 3 is distributed over (2 + 5) and in the right side, 3 is distributed between both terms individually, 2 and 5. So, the application of the distributive property here in the given equation seems correct.

Step 2: Perform Left Side Arithmetic

Let's calculate the left side of the equation i.e., \((2 + 5) * 3 = 21\).

Step 3: Perform Right Side Arithmetic

Now, let's calculate the right side of the equation which is \(2 * 3 + 5 * 3 = 6 + 15 = 21\).

Key Concepts

Arithmetic

Arithmetic involves basic mathematical operations, such as addition, subtraction, multiplication, and division. In the given exercise, we are dealing with addition and multiplication. On the left side of the equation,
we have \((2+5) \cdot 3 \), meaning we add 2 and 5 first, resulting in 7, and then multiply by 3, resulting in 21.
On the right side, the same number 3 is multiplied separately with 2 and 5.
Let's highlight each step:

• First, add 2 and 5 to get 7.
• Multiply 7 by 3 to get 21.
• Or, multiply 2 by 3 and 5 by 3, then add the results to get 6 + 15 = 21.

This clear pattern of arithmetic operations helps in verifying if the distributive property is correctly applied.

Equation Solving

Equation solving is the process of finding the values of variables that satisfy a given mathematical equation. In this instance, we're not trying to find a variable but instead verify the truth of an equation.
This means ensuring that both sides of the equation are equal using arithmetic and properties like the distributive property.
Here's the step-by-step:

• Check if both sides perform the same mathematical operations.
• Verify if each side simplifies to the same result (21 in this case).

By solving the left side and right side separately, we can confirm if the equation stands true or false.

Mathematical Proofs

Mathematical proofs involve logical reasoning to demonstrate the truth of a mathematical statement. In the context of the exercise, the proof hinges on the validity of the distributive property.
The distributive property states that multiplying a number by a sum is the same as multiplying each addend individually and then adding the results.

• Proof of the equation \((2 + 5) \cdot 3 = 2 \cdot 3 + 5 \cdot 3\) involves checking each part:
• Compute both sides using the arithmetic process.
• Compare the outcomes to affirm equality, hence proving the statement true.

This structured approach helps in understanding why the operation works universally across similar equations.