Solving For X The Equation 4(x+3)-7(2x-5)=3x+9

Hey everyone! Let's dive into a common algebra problem: solving for the unknown variable, x. In this guide, we're going to break down the equation 4(x+3)-7(2x-5)=3x+9 step-by-step, making it super easy to understand. We’ll not only find the value of x but also explore the underlying concepts and techniques involved. So, grab your pencils and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into the specifics of our equation, let's quickly recap some foundational concepts in algebra. Algebraic equations are mathematical statements that show the equality between two expressions. These expressions contain variables (usually denoted by letters like x, y, or z), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division, etc.). Solving algebraic equations essentially means finding the value(s) of the variable(s) that make the equation true. The most important thing to remember when solving for a variable is that whatever you do to one side of the equation, you must also do to the other side to maintain the balance. This principle ensures that the equality remains intact throughout the solution process. Think of it like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it balanced. In our case, we're dealing with a linear equation, which is an equation where the highest power of the variable is 1. These equations have a straightforward solution process, typically involving isolating the variable on one side of the equation. We’ll use techniques like distribution, combining like terms, and inverse operations to achieve this goal. Mastering these basic principles is crucial for tackling more complex algebraic problems in the future. So, let's keep these concepts in mind as we move forward and dissect the given equation.

Step 1: Distribute the Constants

The first crucial step in solving the equation 4(x+3)-7(2x-5)=3x+9 involves distributing the constants outside the parentheses to the terms inside. This means we need to multiply the 4 by both x and +3 in the first part of the equation, and we need to multiply the -7 by both 2x and -5 in the second part. Distribution is a fundamental technique in algebra that simplifies equations by removing parentheses and combining terms. Let’s break this down: When we distribute the 4 across (x+3), we get 4 * x + 4 * 3, which simplifies to 4x + 12. Remember, the distributive property states that a(b + c) = ab + ac. Similarly, when we distribute the -7 across (2x-5), we need to be careful with the signs. -7 multiplied by 2x gives us -14x, and -7 multiplied by -5 gives us +35. Notice how multiplying two negative numbers results in a positive number. So, the expression -7(2x-5) becomes -14x + 35. Now, let's rewrite the entire equation with these distributed terms: 4x + 12 - 14x + 35 = 3x + 9. By completing this first step, we've eliminated the parentheses and made the equation much easier to work with. This is a significant step towards isolating x and finding its value. So, always remember to look for opportunities to distribute constants as the first move in simplifying algebraic equations.

Step 2: Combine Like Terms

Now that we've distributed the constants in the equation 4(x+3)-7(2x-5)=3x+9, which now looks like 4x + 12 - 14x + 35 = 3x + 9, the next step is to combine like terms. Combining like terms is a critical technique in algebra that simplifies equations by grouping terms that have the same variable and exponent. In this case, we have terms with x and constant terms (numbers without variables) on the left side of the equation. Let's identify the like terms: We have 4x and -14x, which are both terms containing the variable x. We also have the constants 12 and 35. To combine the x terms, we add their coefficients: 4x - 14x = -10x. To combine the constant terms, we simply add them: 12 + 35 = 47. So, on the left side of the equation, we can replace 4x + 12 - 14x + 35 with -10x + 47. The equation now looks like: -10x + 47 = 3x + 9. This simplified form is much easier to handle. Combining like terms helps to condense the equation, making it clearer and reducing the chance of errors in subsequent steps. This step is crucial for efficiently solving the equation, so make sure to identify and combine like terms whenever possible. By reducing the number of terms, we're one step closer to isolating the variable x and finding its value.

Step 3: Isolate the Variable Terms

With our equation simplified to -10x + 47 = 3x + 9, the next crucial step is to isolate the variable terms on one side of the equation. This means we want to get all the terms containing x on one side and all the constant terms on the other side. To do this, we’ll use inverse operations, which involve performing the opposite operation to both sides of the equation to maintain balance. The goal here is to move the term with x from the right side to the left side. Currently, we have 3x on the right side. To eliminate this term from the right, we subtract 3x from both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. This gives us: -10x + 47 - 3x = 3x + 9 - 3x. On the left side, we combine the x terms: -10x - 3x = -13x. On the right side, the 3x terms cancel each other out, leaving us with just 9. So, the equation now looks like: -13x + 47 = 9. By isolating the variable terms, we've taken a significant step towards solving for x. This process sets us up for the next step, where we'll isolate the variable x completely by dealing with the remaining constant term. So, remember the importance of using inverse operations to move variable terms around while maintaining the equation's balance.

Step 4: Isolate the Constant Terms

Following our progress in isolating the variable terms, we've arrived at the equation -13x + 47 = 9. Now, our focus shifts to isolating the constant terms on the other side of the equation. This means we want to move the constant term (47) from the left side to the right side. To achieve this, we again employ the principle of inverse operations. Since 47 is being added to -13x, we perform the inverse operation by subtracting 47 from both sides of the equation. This ensures we maintain the balance and equality. Subtracting 47 from both sides gives us: -13x + 47 - 47 = 9 - 47. On the left side, the +47 and -47 cancel each other out, leaving us with just -13x. On the right side, we perform the subtraction: 9 - 47 = -38. So, the equation now simplifies to: -13x = -38. By isolating the constant terms, we've effectively separated the variable term from the constant terms, bringing us closer to the final solution. This step is crucial because it sets up the final operation needed to solve for x, which involves dividing both sides by the coefficient of x. Remember, the key to isolating terms is to use inverse operations strategically, ensuring that whatever operation you perform on one side of the equation, you also perform on the other side. With the constant terms now isolated, we're just one step away from finding the value of x.

Step 5: Solve for x

Having successfully isolated both the variable and constant terms, we've arrived at the simplified equation -13x = -38. The final step in our journey is to solve for x itself. This involves getting x completely alone on one side of the equation. Currently, x is being multiplied by -13. To undo this multiplication and isolate x, we need to perform the inverse operation, which is division. We divide both sides of the equation by -13. This ensures that we maintain the balance and equality of the equation. Dividing both sides by -13, we get: (-13x) / -13 = -38 / -13. On the left side, the -13 in the numerator and the -13 in the denominator cancel each other out, leaving us with just x. On the right side, we divide -38 by -13. A negative number divided by a negative number results in a positive number. So, -38 / -13 simplifies to 38/13. Therefore, the value of x is 38/13. We can express this as an improper fraction or convert it to a mixed number if desired. However, 38/13 is the exact solution. By completing this final step, we've successfully solved for x in the equation. This process demonstrates the power of using inverse operations to isolate the variable and find its value. Always remember to check your solution by substituting it back into the original equation to ensure it holds true. Congratulations, you've mastered solving this algebraic equation!

Conclusion

Alright, guys! We've made it to the end! We successfully navigated through the equation 4(x+3)-7(2x-5)=3x+9 and found that x equals 38/13. We've covered everything from distributing constants to isolating terms and finally solving for x. Remember, practice makes perfect, so keep tackling those algebraic challenges! Understanding these steps not only helps in solving similar equations but also builds a solid foundation for more advanced mathematical concepts. So, keep practicing, keep exploring, and never stop learning. You've got this!