Solving Equations: First Step Explained

Let's dive into how to solve equations, specifically focusing on the very first step. We'll use the equation $3p - 7 + p = 13$ as our example. Understanding this initial step is crucial because it sets the foundation for finding the value of the variable. So, let’s break it down in a way that’s super easy to grasp!

Understanding the Equation

Before we jump into solving, let's quickly understand what the equation $3p - 7 + p = 13$ is all about. In this equation, $p$ is our variable, which means it's the unknown value we're trying to find. The equation tells us that if we multiply this unknown value by 3, subtract 7, and then add the unknown value again, the result should be 13. Our mission is to isolate $p$ on one side of the equation to figure out what its value is. Think of it like a puzzle where we need to rearrange the pieces to reveal the hidden number.

Combining Like Terms: The Key First Step

The most efficient first step in solving the equation $3p - 7 + p = 13$ involves combining like terms. What exactly are "like terms"? They are terms in the equation that contain the same variable raised to the same power. In our equation, we have two terms that contain the variable $p$: $3p$ and $+p$. We can combine these two terms because they both involve $p$ to the first power. To combine them, we simply add their coefficients. The coefficient of $3p$ is 3, and the coefficient of $+p$ is 1 (since $p$ is the same as $1p$). So, we add 3 and 1 to get 4. This means we can replace $3p + p$ with $4p$. The equation then becomes $4p - 7 = 13$.

Why Combine Like Terms?

Combining like terms is essential because it simplifies the equation. A simpler equation is easier to solve! By combining $3p$ and $p$ into $4p$, we reduce the number of terms in the equation, making it more manageable. This step doesn't change the value of the equation; it just rewrites it in a more concise form. It's like tidying up before you start a project – it makes the whole process smoother and less confusing. Trust me, guys, this little trick saves a lot of headaches down the road!

Let's Look at the Options

Now, let's revisit the original question. We were given the equation $3p - 7 + p = 13$ and asked what the resulting equation is after the first step. We've established that the best first step is to combine like terms, which results in $4p - 7 = 13$. So, let's examine the answer choices provided:

• A. $p - 7 = 13 - 3p$: This option is incorrect. It seems like someone tried to move terms around, but it's not the correct first step and doesn't simplify the equation in a helpful way.
• B. $2p - 7 = 13$: This option is also incorrect. It looks like there might have been a mistake in combining the $p$ terms. Remember, $3p + p$ should be $4p$, not $2p$.
• C. $3p - 7 = 13 - p$: This option is incorrect for similar reasons as option A. It involves rearranging terms instead of combining like terms.
• D. $4p - 7 = 13$: This is the correct answer! As we discussed, combining the like terms $3p$ and $p$ gives us $4p$, resulting in the equation $4p - 7 = 13$.

Why This First Step Matters

Okay, so we've found the correct first step. But why does it even matter? Well, think of solving an equation like building a house. The first step is laying the foundation. If your foundation is shaky, the rest of the house won't be stable. Similarly, if you don't start solving the equation correctly, you're more likely to make mistakes later on. Combining like terms is a fundamental algebraic skill that simplifies equations and sets you up for success in solving them accurately.

Avoiding Common Mistakes

• Not combining like terms: Some people might try to jump straight into moving terms around without simplifying first. This can lead to unnecessary complexity and increase the chance of errors.
• Incorrectly combining like terms: Make sure you're only combining terms that have the same variable raised to the same power. Also, double-check your arithmetic when adding or subtracting the coefficients.
• Forgetting the sign: Pay close attention to the signs (positive or negative) of the terms when combining them. A simple sign error can throw off the entire solution.

Let's Summarize

So, to recap, when solving the equation $3p - 7 + p = 13$, the resulting equation after the first step (combining like terms) is $4p - 7 = 13$. This involves adding the coefficients of the $p$ terms (3 and 1) to get 4, thus simplifying the equation. Mastering this first step sets a solid foundation for accurately solving the equation and finding the value of $p$.

Level Up Your Equation-Solving Skills

Now that you've got the hang of the first step, here are a few tips to further improve your equation-solving abilities:

1. Practice regularly: The more you practice, the more comfortable you'll become with identifying and combining like terms.
2. Double-check your work: Always take a moment to review your steps to catch any potential errors.
3. Break down complex equations: If you encounter a more complicated equation, try to break it down into smaller, more manageable steps.
4. Seek help when needed: Don't be afraid to ask for help from a teacher, tutor, or online resources if you're struggling with a particular concept.

More Examples to try

To solidify your understanding, let's look at a couple more examples:

Example 1:

Solve: $5x + 3 - 2x = 10$

First, combine like terms: $5x - 2x = 3x$

Resulting equation: $3x + 3 = 10$

Example 2:

Solve: $2y - 8 + 4y + 1 = 5$

First, combine like terms: $2y + 4y = 6y$ and $-8 + 1 = -7$

Resulting equation: $6y - 7 = 5$

Why Equations Matter

You might be wondering, "Why are we even learning about equations?" Well, equations are fundamental to many areas of life! They're used in science, engineering, economics, computer programming, and many other fields. Equations allow us to model real-world situations, make predictions, and solve problems. For example, engineers use equations to design bridges and buildings, economists use equations to forecast economic trends, and scientists use equations to describe the laws of nature.

Equations in Everyday Life

Equations aren't just for scientists and mathematicians. They're also used in everyday life, often without us even realizing it! Here are a few examples:

• Cooking: Recipes often involve equations. For example, if you want to double a recipe, you need to double all the ingredients.
• Budgeting: Creating a budget involves equations. You need to make sure that your income is greater than or equal to your expenses.
• Travel: Calculating travel time and distance involves equations. For example, distance = speed × time.

Conclusion

In conclusion, understanding the first step in solving equations – combining like terms – is crucial for success in algebra and beyond. By simplifying equations, we make them easier to solve and reduce the likelihood of errors. So, remember to always combine like terms first, double-check your work, and practice regularly. With these tips, you'll be well on your way to mastering equation-solving and unlocking its many applications in the real world. Keep up the great work, guys, and happy solving!