Problem 7 - Elementary Algebra

Question

Solve the equations and inequalities for the following problems. $$ 3(2 a+4)=2(a+3) $$

Step-by-Step Solution

Answer

Answer: The value of 'a' in the given equation is \(\frac{-3}{2}\).

Step 1: Apply the Distributive Property

Multiply 3 by the terms in the parenthesis: \((2a + 4)\), and 2 by the terms in the parenthesis: \((a + 3)\). This simplifies the equation as: $$ 6a + 12 = 2a + 6 $$

Step 2: Collect Like Terms

Subtract 2a from both sides of the equation to have 'a' terms on one side: $$ 6a - 2a + 12 = 2a - 2a + 6 $$ Which simplifies to: $$ 4a + 12 = 6 $$

Step 3: Isolate Variable 'a'

Subtract 12 from both sides of the equation to isolate the term with the variable 'a': $$ 4a + 12 - 12 = 6 - 12 $$ Simplify the equation: $$ 4a = -6 $$ Now, divide both sides by 4 to get the value of 'a': $$ a = \frac{-6}{4} $$

Step 4: Simplify the Fraction

The fraction can be reduced by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2: $$ a = \frac{-6 \div 2}{4 \div 2} $$ This simplifies to: $$ a = \frac{-3}{2} $$ So, the solution to the equation is \(a = \frac{-3}{2}\).

Key Concepts

Understanding the Distributive Property

The distributive property is often one of the earliest algebraic tools you'll come across, and it’s extremely useful. It essentially tells us how to multiply something outside a parenthesis by every term inside the parenthesis. For example, in the problem, we have the expression \(3(2a + 4)\). To apply the distributive property, you multiply 3 by each term inside the parenthesis:

• 3 times 2a gives you 6a.
• 3 times 4 gives you 12.

So \(3(2a + 4)\) becomes \(6a + 12\). Similarly, distribute the 2 in \(2(a + 3)\):

• 2 times a gives you 2a.
• 2 times 3 gives you 6.

Thus, \(2(a + 3)\) becomes \(2a + 6\). This helps transform the expression into a form that's easier to solve.

Combining Like Terms

Combining like terms means grouping together terms that have the same variable raised to the same power. This is useful for simplifying an equation and making it easier to solve. In our example, after applying the distributive property, the equation turns into \(6a + 12 = 2a + 6\).
To simplify this, we need all the terms with 'a' on one side. Subtract 2a from both sides to collect the 'a' terms together:

• Subtract 2a from 6a to get 4a.

Now, each side of the equation looks like this:
\(4a + 12 = 6\)
Now the equation is simpler to work with since all the identical terms are combined.

Isolating the Variable

After combining like terms, the next step is to isolate the variable. This means getting the variable by itself on one side of the equation. Currently, we have \(4a + 12 = 6\).
To isolate 'a', we need to remove the constant (12) from the left side. So, we subtract 12 from both sides:

• \(4a + 12 - 12 = 6 - 12\)
• Which simplifies to \(4a = -6\)

Now, all that's left is to get 'a' alone. Since 'a' is currently multiplied by 4, divide both sides by 4:

• \(a = \frac{-6}{4}\)

This step of isolating the variable is crucial because it allows you to clearly identify the value of your variable.

Simplifying Fractions

Simplifying fractions is the final step in solving our equation, resulting in a cleaner and neater answer. In the equation \(a = \frac{-6}{4}\), both the numerator and the denominator can be simplified by finding their greatest common divisor (GCD), which is 2.

• Divide -6 by 2 to get -3.
• Divide 4 by 2 to get 2.

So, \(\frac{-6}{4}\) simplifies to \(\frac{-3}{2}\).
Simplifying fractions makes them easier to interpret and can often be a requirement for a correct answer in math exercises. This ensures that your solution is presented in its simplest form, which not only looks better but also communicates clearly to anyone reviewing your work.