Problem 95 - Intermediate Algebra

Question

Solve each equation. $$ -4[p-(3-p)]=3(6 p-2) $$

Step-by-Step Solution

Answer

The solution is \(p = \frac{9}{13}\).

Step 1: Distribute Inside the Parentheses

First, let's simplify the expression inside the parentheses by removing them. The expression inside the brackets is \([p - (3 - p)]\). Distribute the negative sign to both terms in the expression, resulting in the simplified form: \[p - 3 + p = 2p - 3\].

Step 2: Distribute the Outer Expression

Now, distribute -4 to both terms of the expression \(2p - 3\) from Step 1. This gives us:\[-4(2p - 3) = -8p + 12\].

Step 3: Simplify the Right Side

Distribute the 3 on the right-hand side of the equation:\[3(6p - 2) = 18p - 6\].

Step 4: Combine and Solve

Now that both sides of the equation are simplified, we have:\[-8p + 12 = 18p - 6\].Add \(8p\) to both sides to get:\[12 = 26p - 6\].Next, add 6 to both sides:\[18 = 26p\].

Step 5: Isolate the Variable

Finally, divide both sides by 26 to isolate \(p\):\[p = \frac{18}{26}\].Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, 2:\[p = \frac{9}{13}\].

Key Concepts

Distributive Property

The distributive property is like a magic rule in math that helps us simplify expressions with parentheses. When you have something like \(-4[p-(3-p)]\) in an equation, you apply this rule to "spread" the outside number, in this case -4, to each term inside the brackets. It works like this:

• We start with the expression inside: \(p - (3 - p)\).
• Simplify it by distributing the minus sign: \(p - 3 + p = 2p - 3\).
• Next, apply -4 to each term from the expanded expression: \(-4(2p - 3) = -8p + 12\).

By using the distributive property, we effectively eliminate the parentheses, allowing us to move on to solving the equation further. It’s vital for breaking down complex problems into smaller, manageable parts.
It's like taking a big puzzle and dividing it into smaller pieces that are easier to work with.

Combining Like Terms

After distributing, we often need to combine like terms to simplify the equation further. Like terms are terms whose variables (and their exponents, if any) are the same. In our equation, once the distributive property is applied, we focus on combining terms to make the equation neat.

• Consider terms on the left-hand side: \(-8p + 12\).
• And on the right-hand side: \(18p - 6\).
• The aim is to put all the \(p\) terms together and coefficients together separately.

We first tackle the like terms on either side: moving \(8p\) from left to right results in \(8p + 18p = 26p\). Next, handle the constant terms by adding 6 from the right to 12 on the left: \(12 + 6 = 18\).
This simplifies your equation to \(18 = 26p\).
Combining like terms makes it much easier to see what you need to do next to solve for the variable.

Isolating the Variable

This is the part where we focus entirely on solving for the unknown variable, usually represented as a letter, such as \(p\). The goal is to get the variable by itself on one side of the equation.
Isolating the variable involves moving all other numbers to the opposite side:

• In our solved equation: \(18 = 26p\).
• We need \(p\) alone on one side.

Achieve this by dividing both sides by the coefficient of \(p\), which is 26:
\(p = \frac{18}{26}\).
Simplify this fraction by dividing the numerator and the denominator by their greatest common divisor, which is 2: \(p = \frac{9}{13}\). Splitting numbers through division is crucial because it allows the variable to take up its solitary place.
Once isolated, the mystery of the equation is solved, revealing the exact value of the variable.