Problem 16 - Prealgebra

Question

Solve each equation. Check your solution. $$9=15+2 p$$

Step-by-Step Solution

Answer

The solution to the equation is \( p = -3 \).

Step 1: Isolate the variable term

First, we want to isolate the term with the variable on one side of the equation. Our equation is \( 9 = 15 + 2p \). Subtract 15 from both sides to achieve this: \[9 - 15 = 2p\]This simplifies to \[-6 = 2p\].

Step 2: Solve for the variable

Now, we need to solve for \( p \). Since \( 2p = -6 \), we divide both sides of the equation by 2:\[p = \frac{-6}{2}\]Simplifying gives us:\[p = -3\].

Step 3: Check the solution

To verify our solution is correct, substitute \( p = -3 \) back into the original equation \( 9 = 15 + 2p \) and check if both sides are equal.\[9 = 15 + 2(-3)\]Simplify the right-hand side:\[9 = 15 - 6\]\[9 = 9\].Since both sides are equal, the solution is confirmed.

Key Concepts

Solving Equations

Solving equations is like a treasure hunt for math lovers. You're on a quest to find the value of the unknown variable. Simple linear equations, like the one in our exercise, usually follow a straightforward process. First, make sure all the variable terms are on one side and all the constant terms are on the other. This means moving numbers here and there, but remember that whatever you do to one side of the equation, you do to the other as well. It keeps the equation balanced, just like a seesaw. In our example, we started with the equation \(9 = 15 + 2p\), and we moved the 15 by subtracting it from both sides, which is like moving ingredients from one side of a recipe to another. Once you've neatened things up, you can figure out the value of \(p\) or any other variable you're working with.

Checking Solutions

After you think you've solved an equation, it's time to double-check. This is an important step because it confirms your answer is correct. Checking your solution means substituting the found value back into the original equation. If the equation stays true, congrats! You've found the right treasure. For example, plugging \(p = -3\) back into our original equation, we computed \(9 = 15 + 2(-3)\) which simplifies to \(9 = 9\). Both sides agreed, which meant that our solution was spot on. Consistently checking your work helps you avoid mistakes, just like rereading a map ensures you’re on the right path.

Isolating Variables

To isolate a variable means to get it by itself on one side of the equation. This is a crucial step in solving equations because it essentially translates the equation into a statement like 'p equals something'. Imagine the task of isolating a variable as untangling a knot: you must carefully rearrange things until each piece is free and clear. In our example, we needed to isolate \(p\). We did this by first getting rid of the 15 that was up alongside it, moving it to the other side of the equation through subtraction. Then, we cleared out the number \(2\), which was multiplying the \(p\), by dividing it out. The clear message here is that isolating the variable simplifies the equation and makes finding the solution straightforward.