Solving For Equivalent Equations 7x + 20 = 6 A Step By Step Guide
Hey guys! Today, we're diving deep into the world of algebraic equations, specifically tackling the question: Which of these equations is equivalent to 7x + 20 = 6? This might seem like a straightforward math problem, but understanding the underlying principles of equation equivalence is crucial for mastering algebra. We'll break down the original equation, explore the given options, and uncover the step-by-step process to identify the equivalent equation. So, buckle up and let's get started!
Understanding Equivalent Equations
Before we jump into solving, let's quickly define what equivalent equations actually are. Equivalent equations are equations that have the same solution. In simpler terms, if you solve for 'x' in two equivalent equations, you'll get the same value. To determine if equations are equivalent, we need to manipulate them using algebraic operations to see if they can be transformed into each other or if they share the same solution. This involves applying properties like the addition, subtraction, multiplication, and division properties of equality. These properties allow us to add, subtract, multiply, or divide both sides of an equation by the same number without changing the solution. Mastering these properties is key to solving various algebraic problems. Think of it like this: you're trying to balance a scale. Whatever you do to one side, you must do to the other to maintain the balance. This concept is the foundation of solving equations.
The goal is to isolate the variable, in this case 'x', on one side of the equation. To achieve this, we often perform inverse operations. For example, if an equation involves addition, we use subtraction to undo it. If it involves multiplication, we use division. It's like a dance – each step is carefully chosen to bring us closer to the solution. Remember, the journey of solving equations is not just about finding the answer but also about understanding why the steps we take are valid. This understanding will empower you to tackle more complex problems in the future. So, let's keep this in mind as we move forward and dissect our problem step-by-step.
Analyzing the Original Equation: 7x + 20 = 6
Our starting point is the equation 7x + 20 = 6. To find equivalent equations, we first need to understand the solution to this equation. This understanding acts as our benchmark – any equivalent equation must have the same solution. Let's walk through the steps to solve for 'x'. The first step is to isolate the term with 'x'. We do this by subtracting 20 from both sides of the equation. This is a direct application of the subtraction property of equality. So, we have:
7x + 20 - 20 = 6 - 20
This simplifies to:
7x = -14
Now, 'x' is being multiplied by 7. To isolate 'x', we need to perform the inverse operation, which is division. We divide both sides of the equation by 7, using the division property of equality:
7x / 7 = -14 / 7
This gives us:
x = -2
So, the solution to the original equation 7x + 20 = 6 is x = -2. This means any equation equivalent to the original must also have the solution x = -2. This is our golden rule, our guiding star as we navigate the options presented to us. We've established our benchmark, and now we're ready to test the waters. Let's dive into the options and see which one holds true to our solution.
Evaluating Option A: 7x - 12 = 2
Let's investigate the first option, A. 7x - 12 = 2. To determine if this equation is equivalent to our original equation, we need to solve for 'x' and see if we get the same solution, which we know is x = -2. The first step in solving this equation is to isolate the term with 'x'. We can do this by adding 12 to both sides of the equation. Remember, we're applying the addition property of equality – what we do to one side, we do to the other:
7x - 12 + 12 = 2 + 12
This simplifies to:
7x = 14
Now, to isolate 'x', we divide both sides of the equation by 7, using the division property of equality:
7x / 7 = 14 / 7
This gives us:
x = 2
Notice something crucial here. The solution to equation A is x = 2, but the solution to our original equation is x = -2. These solutions are different! This tells us definitively that option A, 7x - 12 = 2, is not equivalent to our original equation 7x + 20 = 6. It's like trying to fit a square peg in a round hole – it just doesn't work. We've successfully eliminated one option. Let's move on to the next and continue our quest for the equivalent equation.
Analyzing Option B: 3x - 5 = 11
Now, let's turn our attention to option B. 3x - 5 = 11. Just like with option A, our mission is to solve for 'x' and see if the solution matches our benchmark solution of x = -2 from the original equation. To begin, we need to isolate the term containing 'x'. We can accomplish this by adding 5 to both sides of the equation. This is the trusty addition property of equality in action:
3x - 5 + 5 = 11 + 5
This simplifies to:
3x = 16
Next, to get 'x' all by itself, we'll divide both sides of the equation by 3, using the division property of equality:
3x / 3 = 16 / 3
This gives us:
x = 16/3
Or, approximately x = 5.33. Take a close look at this result. The solution we've found for equation B is x = 16/3, which is clearly not equal to our original solution of x = -2. This is a crucial observation! It tells us, without a doubt, that option B, 3x - 5 = 11, is not an equivalent equation to 7x + 20 = 6. We've successfully crossed off another option from our list. Only two remain. Let's keep our momentum going and investigate option C.
Scrutinizing Option C: 10 - 3x = 16
Let's shift our focus to option C. 10 - 3x = 16. Our consistent strategy remains the same: solve for 'x' and compare the solution to our original benchmark of x = -2. To isolate the term with 'x', which is -3x, we first need to get rid of the 10 on the left side. We can do this by subtracting 10 from both sides of the equation. This is, once again, a straightforward application of the subtraction property of equality:
10 - 3x - 10 = 16 - 10
This simplifies to:
-3x = 6
Now, we have -3 multiplied by 'x'. To isolate 'x', we need to divide both sides of the equation by -3. Remember, we're still adhering to the division property of equality:
-3x / -3 = 6 / -3
This gives us:
x = -2
Eureka! Look what we've found. The solution to equation C is x = -2. This perfectly matches the solution of our original equation, 7x + 20 = 6. This is a strong indication that option C is indeed an equivalent equation. But to be absolutely sure, and to complete our analysis, let's take a quick look at the final option. It's always good to be thorough in mathematics!
Examining Option D: 15 + 2x = 19
Finally, we arrive at option D. 15 + 2x = 19. Following our established pattern, we need to solve for 'x' and compare the result to our original solution of x = -2. To isolate the term with 'x', which is 2x, we first subtract 15 from both sides of the equation, employing the subtraction property of equality:
15 + 2x - 15 = 19 - 15
This simplifies to:
2x = 4
Now, to isolate 'x', we divide both sides of the equation by 2, applying the division property of equality:
2x / 2 = 4 / 2
This gives us:
x = 2
Ah, and here we have it. The solution to equation D is x = 2, which is not the same as our benchmark solution of x = -2. This confirms that option D, 15 + 2x = 19, is not equivalent to the original equation 7x + 20 = 6. We've now thoroughly examined all the options.
The Verdict: Option C is the Equivalent Equation
After a comprehensive analysis of all the options, we've definitively identified the equivalent equation. Through a step-by-step process of solving for 'x' in each equation and comparing the solutions, we've discovered that:
- Option A (7x - 12 = 2) has a solution of x = 2 (Not equivalent)
- Option B (3x - 5 = 11) has a solution of x = 16/3 (Not equivalent)
- Option C (10 - 3x = 16) has a solution of x = -2 (Equivalent!)
- Option D (15 + 2x = 19) has a solution of x = 2 (Not equivalent)
Therefore, the correct answer is C. 10 - 3x = 16. This equation shares the same solution (x = -2) as our original equation, 7x + 20 = 6, making them equivalent. We've successfully navigated the world of algebraic equations and pinpointed the equivalent one. Remember, guys, the key to solving these problems lies in understanding the properties of equality and applying them systematically. Keep practicing, and you'll become equation-solving masters!
Final Thoughts on Equation Equivalence
Understanding equivalent equations is not just about solving specific problems; it's about building a strong foundation in algebraic thinking. The principles we've discussed today – the properties of equality, the process of isolating variables, and the importance of checking solutions – are all fundamental to more advanced mathematical concepts. As you continue your mathematical journey, you'll encounter more complex equations and systems of equations. The skills you've honed here will serve you well. So, keep exploring, keep questioning, and keep practicing. Mathematics is a journey of discovery, and each problem you solve is a step forward. You got this!