Solving Equations Step-by-Step Justifying Each Property Used

Hey guys! Ever feel lost in the maze of solving equations? It's like trying to find your way through a dense forest, but don't worry, I'm here to be your guide. We're going to break down each step, showing not just how but why we do what we do. We'll justify every move using the properties that make it legit. Let's dive into an example and make equation-solving a piece of cake!

The Equation at Hand

We're going to tackle this equation: $2(10-13x) + 9x = -34x + 60$.

Step 1: $20 - 26x + 9x = -34x + 60$

In this initial step, we're using the Distributive Property. Think of it like sharing the love (or multiplication) to everyone inside the parentheses. The Distributive Property states that a(b + c) = ab + ac. So, we multiply the 2 outside the parentheses by both the 10 and the -13x inside.

• What we did: We distributed the 2 across the terms inside the parenthesis.
• How it looks: $2 * 10 = 20$ and $2 * -13x = -26x$. This gives us $20 - 26x + 9x = -34x + 60$.

The Distributive Property is super important because it helps us simplify expressions by getting rid of parentheses. Without it, we'd be stuck trying to solve an equation with a bunch of terms trapped inside! You see, the Distributive Property makes the equation more workable and sets the stage for the next steps. It's like unlocking a door that was previously closed, allowing us to move forward in our problem-solving journey. In the grand scheme of algebra, mastering the Distributive Property is a foundational skill. It pops up everywhere, from simple equations to more complex algebraic manipulations. So, nailing this concept is a huge win for your math toolkit!

Step 2: $-17x + 20 = -34x + 60$

Here, we're all about keeping things tidy by Combining Like Terms. Like terms are terms that have the same variable raised to the same power. In our case, we have $-26x$ and $+9x$ on the left side of the equation. Think of it like sorting your socks – you put the pairs together, right? We're doing the same with these terms.

• What we did: We combined the 'x' terms on the left side.
• How it looks: $-26x + 9x$ combines to $-17x$. So, our equation becomes $-17x + 20 = -34x + 60$.

Combining like terms is a crucial step because it simplifies the equation, making it easier to handle. Imagine trying to solve an equation with a dozen 'x' terms scattered everywhere – it would be a nightmare! By bringing them together, we reduce the clutter and create a clearer path to the solution. It's like decluttering your workspace before starting a project; a clean space helps you think more clearly and work more efficiently. This step highlights the importance of organization in mathematics. Just as a well-organized essay is easier to read, a well-organized equation is easier to solve. By combining like terms, we're essentially organizing the equation, setting ourselves up for success in the subsequent steps.

Step 3: $-17x + 34x = 60 - 20$

Now, we're employing the Addition Property of Equality and the Subtraction Property of Equality. These properties state that you can add or subtract the same value from both sides of an equation without changing its solution. Think of it like a balanced scale – if you add or remove the same weight from both sides, it stays balanced.

• What we did: We added $34x$ to both sides and subtracted $20$ from both sides.
• How it looks: Adding $34x$ to both sides of $-17x + 20 = -34x + 60$ cancels out the $-34x$ on the right, and subtracting 20 from both sides isolates the x terms on the left and the constants on the right. This gives us $17x = 40$.

The Addition and Subtraction Properties of Equality are fundamental tools in solving equations. They allow us to move terms around while maintaining the equation's balance, which is crucial for isolating the variable we're trying to solve for. Without these properties, we'd be stuck with variables and constants mixed up on both sides, making it impossible to find a solution. It's like having a toolbox with all the right wrenches – these properties are the wrenches we need to loosen and tighten the equation to get it into the right form. By using these properties, we're strategically manipulating the equation to get closer to our goal of isolating 'x'.

Step 4: x = rac{40}{17}

Finally, we use the Division Property of Equality. This property states that you can divide both sides of an equation by the same non-zero value without changing the solution. We're using it to isolate 'x' completely.

• What we did: We divided both sides of $17x = 40$ by 17.
• How it looks: Dividing both sides by 17 gives us x = rac{40}{17}.

The Division Property of Equality is the final step in our journey to solve for 'x'. It's like the last piece of the puzzle, allowing us to reveal the value of the variable. Without this property, we'd be stuck with 'x' multiplied by a coefficient, unable to determine its true value. This step underscores the importance of inverse operations in solving equations. Division is the inverse operation of multiplication, and by using it, we effectively undo the multiplication that was binding 'x'. The result, x = rac{40}{17}, is the solution to the equation. It's the value that, when plugged back into the original equation, makes the equation true.

Wrapping Up

So, there you have it! We've solved the equation step-by-step, justifying each move with the properties that make it mathematically sound. Remember, solving equations isn't just about getting the right answer; it's about understanding why the steps work. By mastering these properties, you'll be able to tackle any equation that comes your way. Keep practicing, and you'll become an equation-solving pro in no time! You've got this!