Problem 56 - Algebra 2

Question

Solve each equation. Check your solution. \(7 p-4=3(4+5 p)\)

Step-by-Step Solution

Answer

The solution is \( p = -2 \).

Step 1: Expand the Equation

First, we expand the right side of the equation by using the distributive property. The equation is:\[ 7p - 4 = 3(4 + 5p) \]Distributing the 3 we get:\[ 3 \times 4 + 3 \times 5p = 12 + 15p \]So the equation becomes:\[ 7p - 4 = 12 + 15p \]

Step 2: Isolate Terms Involving 'p'

Now, let's move all terms involving \( p \) to one side and constants to the other. Subtract \( 15p \) from both sides:\[ 7p - 15p - 4 = 12 \]Simplify the equation:\[ -8p - 4 = 12 \]

Step 3: Isolate the Term

Add 4 to both sides of the equation to isolate the term involving \( p \):\[ -8p = 16 \]

Step 4: Solve for 'p'

Now, divide both sides by \(-8\) to solve for \( p \):\[ p = \frac{16}{-8} \]Simplify the division:\[ p = -2 \]

Step 5: Verify the Solution

Substitute \( p = -2 \) back into the original equation to check the solution:\[ 7(-2) - 4 = 3(4 + 5(-2)) \]Calculate each side:\[ -14 - 4 = 3(4 - 10) \]\[ -18 = 3(-6) \]Both sides equal \(-18\), hence, \( p = -2 \) is the correct solution.

Key Concepts

Distributive Property

When we encounter an equation like \(7p - 4 = 3(4 + 5p)\), the distributive property is our friend. This property allows us to multiply a single term by each term inside a parenthesis. It’s like distributing gifts to everyone in the room one by one. Here's how it works in our example:

• Take the number 3 from outside the parenthesis.
• Multiply it by each term inside: 3 times 4 and 3 times 5p.

This gives us \(3 \times 4 + 3 \times 5p\), which simplifies to \(12 + 15p\).
So, after using the distributive property, the equation transforms into \(7p - 4 = 12 + 15p\).

This method is crucial for breaking down expressions and simplifying them into manageable pieces.

• Remember: You multiply each term inside by the number outside.
• The property helps simplify complex expressions for easier manipulation.

Isolation of Variables

Once we have simplified an equation using the distributive property, we move on to isolating the variable. In our equation, this involves arranging the terms so that the variable "p" sits alone on one side. This step is essential because it helps us find the exact value of the variable.Starting from \(7p - 4 = 12 + 15p\), we want "p" on one side and numbers on the other. We subtract \(15p\) from both sides to manage the variable terms:

• \(7p - 15p - 4 = 12\).
• Simplify it to \(-8p - 4 = 12\).

The goal is to isolate "p" entirely. To do so, we eliminate constant terms by adding 4 to both sides:

• \(-8p = 16\).

Each algebraic maneuver simplifies our journey to find "p”. By dividing both sides by \(-8\), we retrieve \(p = -2\).
Isolation sets the stage for uncovering the solution through step-by-step simplification.

Solution Verification

Verification is the practice of double-checking our solution to confirm its accuracy. It's like proofreading an important letter before sending it out. In the context of algebra, it involves substituting the solution back into the original equation to see if it holds true. For our solution \(p = -2\):Substitute \(-2\) into the original equation:

• \(7(-2) - 4 = 3(4 + 5(-2))\).
• Simplify the left side: \(-14 - 4 = -18\).
• Simplify the right side: \(3(4 - 10) = 3(-6) = -18\).

Since both sides equal \(-18\), our solution is verified.

Verification ensures that our problem-solving method and calculations are flawless. It gives us confidence that the answer is indeed correct.

• It's a good habit to routinely verify solutions to avoid errors.
• Verification acts as the final check for our mathematical journey.