Problem 3 - Algebra Form and Function

Question

Solve the equations. $$ 2(a+3)=10 $$

Step-by-Step Solution

Answer

Answer: The value of 'a' in the equation is 2.

Step 1: Distribute 2 into parenthesis

To start, we will distribute the 2 into the parenthesis: $$ 2(a+3) = 2a + 6 $$

Step 2: Equalize with the given equation

Now, equalize the simplified expression with the given equation: $$ 2a + 6 = 10 $$

Step 3: Isolate variable 'a'

To solve for 'a', subtract 6 from both sides of the equation: $$ 2a = 4 $$

Step 4: Solve for 'a'

Now, divide by 2 on both sides to find the value of 'a': $$ a = 2 $$ The solution to the equation $$2(a+3) = 10$$ is $$a = 2$$.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that simplifies expressions and makes solving equations easier. It involves spreading a multiplier over each term within the parenthesis. In our example, we have the equation \( 2(a + 3) \). Here, the number 2 is distributed to both \( a \) and 3. This is done as follows:

• Multiply 2 by \( a \) to get \( 2a \).
• Multiply 2 by 3 to get 6.

Doing this distribution, the expression \( 2(a + 3) \) simplifies to \( 2a + 6 \). This step is crucial because it helps break down equations into simpler parts, making it easier to solve for variables. Understanding this property allows you to handle more complex expressions with ease. It’s like unpacking a box—distributing everything inside to see what you have to work with.

Isolating Variables

Isolating variables is a method used to get the variable you need by itself on one side of the equation. This helps you find its value. In our exercise, once we applied the distributive property, the equation became \( 2a + 6 = 10 \).
To isolate \( a \), we need to eliminate any numbers on its side.
In this case:

• Subtract 6 from both sides to keep the equation balanced. This results in:

\[ 2a + 6 - 6 = 10 - 6 \]
So, we have \( 2a = 4 \).
By isolating \( a \), the equation is now simpler, focusing only on what \( a \) equals. Every operation you do to one side of the equation, you also do to the other side. This helps maintain the balance and equality of the equation.

Solving Equations

The final step in solving linear equations involves determining the actual numeric value of the variable. After isolating \( a \) in our simplified equation \( 2a = 4 \), solving it becomes straightforward.

• To remove the number next to \( a \), divide both sides by 2. This keeps the equation balanced:

\[ \frac{2a}{2} = \frac{4}{2} \]
You then find:\( a = 2 \). Solving equations is like solving a mystery where you uncover the value of the variable step by step.
Each action simplifies the equation further, making it easy to find the final answer.
Understanding these principles will help you with similar problems in the future, building a foundation for more complex algebraic processes.