Problem 24 - Prealgebra

Question

Solve each equation. Check your solution. $$13+\frac{p}{3}=-4$$

Step-by-Step Solution

Answer

The solution is \( p = -51 \).

Step 1: Isolate the Fraction Term

First, we need to isolate the term \( \frac{p}{3} \) on one side of the equation. To do that, subtract 13 from both sides of the equation.\[ 13 + \frac{p}{3} - 13 = -4 - 13 \]This simplifies to:\[ \frac{p}{3} = -17 \]

Step 2: Clear the Fraction

Now that the fraction term is isolated, we need to clear the fraction. Multiply both sides of the equation by 3 to solve for \( p \).\[ 3 \cdot \frac{p}{3} = -17 \cdot 3 \]This simplifies to:\[ p = -51 \]

Step 3: Check the Solution

Substitute \( p = -51 \) back into the original equation to ensure that it satisfies the equation.The original equation is:\[ 13 + \frac{p}{3} = -4 \]Replace \( p \) with \(-51\):\[ 13 + \frac{-51}{3} = -4 \]Simplify \( \frac{-51}{3} \) to \(-17\):\[ 13 - 17 = -4 \]Since \( -4 = -4 \), the solution \( p = -51 \) is correct.

Key Concepts

Isolate the Variable

When solving an equation, one of the primary goals is to isolate the variable, which means getting the variable on one side of the equation by itself. This is a crucial step because it simplifies the problem and allows us to find the value of the variable easily. In the given exercise, we're dealing with the term \( \frac{p}{3} \). To isolate this fraction, we need to eliminate anything added or subtracted around it.
Here's how it works:

• Identify the term you need to isolate. Here, it's \( \frac{p}{3} \).
• Look at other terms added or subtracted on the same side (which in this case is '13').
• Subtract 13 from both sides of the equation to remove it, which gives us: \( 13 + \frac{p}{3} - 13 = -4 - 13 \).
  

This process leaves us with \( \frac{p}{3} = -17 \). The variable is now isolated within the fraction, ready for the next step.

Clear the Fraction

After isolating the variable within a fraction, the next logical step is to clear the fraction. Clearing the fraction means transforming the equation in such a way that the variable is no longer in a fractional form. This makes the equation easier to solve.In the step-by-step solution, the fraction \( \frac{p}{3} \) means 'p divided by 3'. To clear this, do the opposite operation of division, which is multiplication:

• Multiply both sides of the equation by 3 (the denominator of the fraction): \( 3 \cdot \frac{p}{3} = -17 \cdot 3 \).
• This multiplication action cancels out the division by 3 on the left side, simplifying the equation to \( p = -51 \).
  

Clearing fractions is crucial because it transforms complicated fractional terms into more manageable whole number equations, leading you straight to finding the value of the variable.

Check the Solution

Once you have found a solution to your equation, it's vital to check whether it holds true when substituted back into the original equation. This step ensures that no mistakes were made during solving.
Here’s how to verify your solution:

• Begin with the original equation: \( 13 + \frac{p}{3} = -4 \).
• Substitute your solution, \( p = -51 \), back into the equation: \( 13 + \frac{-51}{3} = -4 \).
• When substituting, simplify \( \frac{-51}{3} \) to get \(-17 \), so the equation becomes \( 13 - 17 = -4 \).
• Check that both sides of the equation equal: since \(-4 = -4\), the solution is correct.
  

Checking your solution is a critical aspect of solving equations. It's a simple way to confirm your answer is valid and that the problem has been accurately solved.