Problem 19 - Algebra 1

Question

LOGICAL REASONING Decide whether the statement is true or false If false, rewrite the right-hand side of the equation so the statement is true. $$ 3(2+7) \stackrel{?}{=} 3(2)+7 $$

Step-by-Step Solution

Answer

The statement \(3(2+7) = 3(2)+3*7\) is true. No modification is needed for the right-hand side of the equation.

Step 1: Understand the problem

The equation presented is \(3(2+7) \stackrel{?}{=} 3(2)+7\). The objective is to evaluate whether the equation is true or false. If it is false, then modify the right side of the equation to make the statement true.

Step 2: Apply Distributive Property

First, simplify the expression on the left side of the equation by executing the operation inside the parentheses and multiplying by 3. This gives \(3 * 9 = 27\). Then for the right side of the equation, distribute the 3 to both terms inside the parentheses, giving \(3*2 + 3*7 = 6 + 21 = 27\). Thus the equation reads \(27 = 27\).

Step 3: Compare Both Sides

Finally, compare the simplified results on both sides of the equation. We have 27 on both sides which shows that the initial equation holds true. Therefore, there is no need to rewrite the right-hand side of the equation.

Key Concepts

Distributive Property

The distributive property is a powerful tool in algebra that allows us to simplify expressions and solve equations more efficiently. Imagine you have the expression \(3(2+7)\). The distributive property tells us to distribute, or apply, the number outside the parentheses (in this case, 3) to each term inside the parentheses.
This means you'll multiply 3 by both 2 and 7.

• The multiplication calculation \(3 \times 2\) gives us 6.
• The multiplication calculation \(3 \times 7\) results in 21.

Adding the results from these multiplications, we find that \(6 + 21 = 27\).
This allows us to see how expressions can be broken down into simpler parts, making calculations easier, especially when dealing with more complex problems.

Equation Evaluation

Evaluating an equation involves determining whether both sides of an equation are equal after performing all necessary calculations. In our example, we started with the equation \(3(2+7) = 3(2)+7\) and aimed to see if both sides equaled each other.
First, calculate the left-hand side of the equation:

• Simplify inside the parentheses: \(2+7 = 9\).
• Next, multiply by 3, so \(3 \times 9 = 27\).

Now, let's evaluate the right-hand side:

• Again, using the distributive property, distribute 3 to both 2 and 7.
• Calculate: \(3 \times 2 = 6\) and \(3 \times 7 = 21\).
• Finally, add those results: \(6 + 21 = 27\).

Since the left and right sides both equal 27, the equation is balanced, proving the statement true.

Simplification

Simplification is about making expressions easier to work with by reducing them to their simplest form. It's a key skill in solving equations and performing calculations efficiently. In our scenario, simplification was applied both inside the parentheses and during distribution.
Here’s how it works:

• Inside the parentheses, simplify by adding \(2+7\) to get 9 on the left-hand side.
• Then calculate the multiplication \(3 \times 9 = 27\).
• On the right side, use the distributive property to simplify: \(3 \times 2 + 3 \times 7 = 6 + 21 = 27\).

By simplifying both sides of the equation, we ensure we are comparing apples to apples.
Thus, simplifying expressions is a crucial step in verifying the equality of equations and solving for unknowns.