Problem 39 - Algebra 1

Question

Solve the equation if possible. $$ \frac{1}{2}(12 n-4)=14-10 n $$

Step-by-Step Solution

Answer

The solution to the equation is \( n = \frac{3}{4} \).

Step 1: Simplify the equation

First, distribute the \(\frac{1}{2}\) inside the parenthesis to simplify it into a single term: \(\frac{1}{2} * 12n - \frac{1}{2} * 4\) which simplifies to \(6n - 2\). The equation now is \(6n - 2 = 14 - 10n\).

Step 2: Rearrange terms in equation

Now, send -10n to the left side of the equation and -2 to the right side to bring similar terms together. Adding 10n on both sides we have \(6n + 10n = 14 - 2\). Combining like terms we get \(16n = 12\).

Step 3: Solve for the variable

Lastly, solve for variable 'n'. When 16n = 12, n = \(\frac{12}{16}\). Simplifying the right hand side by reducing the fraction, n = \(\frac{3}{4}\).

Key Concepts

Simplification

When solving equations, the first step often involves simplification. Simplification makes an equation easier to handle. In our example, we start with \( \frac{1}{2}(12n - 4) = 14 - 10n \). To simplify this, we apply the fraction \( \frac{1}{2} \) to both terms inside the parentheses. This means multiplying \( \frac{1}{2} \) by each term, resulting in the equation \( 6n - 2 = 14 - 10n \). Simplification helps to reduce complex expressions into simpler forms, making them easier to manipulate in subsequent steps.

Distributive Property

The distributive property is a key concept used in simplification. It states that a(b + c) = ab + ac. This property lets us expand expressions across addition or subtraction inside parentheses. In our equation \( \frac{1}{2}(12n - 4) \), we distribute \( \frac{1}{2} \) to both \( 12n \) and \( -4 \):

• \( \frac{1}{2} \times 12n = 6n \)
• \( \frac{1}{2} \times (-4) = -2 \)

Applying the distributive property in this manner helps us to break down the equation into manageable terms to work with in later steps.

Combining Like Terms

After applying the distributive property, we next tackle combining like terms. Like terms are terms in an equation that have the same variable raised to the same power, allowing them to be combined. In \( 6n + 10n = 14 - 2 \), 'n' terms are like terms. We combine them by adding their coefficients:

• \( 6n + 10n = 16n \)

Combining like terms simplifies an equation further and is a significant step towards isolating the variable.

Rearranging Terms

Rearranging terms involves shifting terms from one side of the equation to the other. This process helps in collecting similar terms together. In our example, we need to move \(-10n\) from the right side so it can combine with \(6n\) already present on the left:

• Add \(10n\) to both sides to get: \( 6n + 10n = 14 - 2 \).
• Move \(-2\) from the left to the right by adding 2 to both sides:
• \(6n + 10n = 14 + 2 \)

By rearranging, we ensure that equivalent expressions remain balanced while isolating terms effectively.

Solving for Variables

Finally, solving for variables involves isolating the variable on one side of the equation. Once similar terms are combined and rearranged, we have \(16n = 12\). To find 'n', divide both sides by 16 to keep the equation balanced:

• \( n = \frac{12}{16} \)

Once you have \(\frac{12}{16}\), reduce the fraction to obtain the simplest form of 'n':

• \( n = \frac{3}{4} \)

Solving for the variable fundamentally concludes the equation-solving process by pinpointing the value of the variable in question.