Equation Solution: Which Equation Solves For -7?

\nLet's dive into the world of equations and solutions! This article will explore how to determine which equation, from a given set, has a specific number as its solution. In this case, we're looking for the equation where substituting -7 for x makes the equation true. This is a fundamental concept in algebra, and understanding it is crucial for solving more complex mathematical problems. We will walk through each option step-by-step, showing how to substitute and simplify to find the correct answer. So, let’s get started and unravel this equation mystery together!

Understanding Equations and Solutions

Before we jump into solving the specific problem, let's make sure we're all on the same page about what equations and solutions actually are. An equation is a mathematical statement that two expressions are equal. It's like a balanced scale, with the left side having the same weight as the right side. This balance is maintained by an equals sign (=).

Think of an equation like this: 2 + 3 = 5. The expression 2 + 3 is on one side, and the expression 5 is on the other. The equals sign tells us that these two expressions have the same value. Simple enough, right?

Now, what's a solution? A solution to an equation is a value (or values) that, when substituted for the variable (usually represented by a letter like x), makes the equation true. In other words, it's the number that makes both sides of the equation balance.

For example, let's look at the equation x + 2 = 5. What value of x would make this equation true? If you guessed 3, you're correct! When we substitute 3 for x, we get 3 + 2 = 5, which is a true statement. Therefore, 3 is the solution to this equation.

Finding solutions is a core skill in algebra. It allows us to solve real-world problems by representing them mathematically. Whether you're calculating the cost of items, determining distances, or even modeling complex systems, equations are your tools, and solutions are the answers you seek. The process often involves isolating the variable on one side of the equation by performing operations on both sides to maintain the balance.

In the following sections, we’ll apply this understanding to the given problem. We will methodically test each equation to see if -7 is indeed the solution, solidifying your grasp on this critical algebraic concept. Remember, the key is substitution and simplification – replacing the variable with the given value and then carefully performing the arithmetic to check for equality.

Analyzing the Equations

Now, let's get to the heart of the problem. We are presented with four equations, and our task is to identify which one has -7 as a solution. To do this, we will substitute -7 for x in each equation and see if the equation holds true. This is a straightforward process of substitution and simplification, but accuracy is key. A small arithmetic error can lead to the wrong conclusion, so we will proceed carefully and methodically.

It’s important to remember the order of operations (often remembered by the acronym PEMDAS or BODMAS) when simplifying. This ensures that we perform calculations in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Let’s list the equations for clarity:

A. 5x + 11 = -24 B. 4x - 23 = 5 C. -x + 3 = -4 D. -3x - 8 = -29

Our strategy is to take each equation, replace x with -7, and then simplify both sides of the equation. If the left side equals the right side after simplification, then -7 is a solution to that equation. If the sides are not equal, then -7 is not a solution.

This process might seem repetitive, but it's a fundamental technique in algebra. It reinforces the concept of what a solution means and how to verify it. Furthermore, it provides practice in arithmetic and algebraic manipulation, skills that are essential for more advanced mathematical studies.

In the following sections, we will analyze each equation individually, step by step. We will show the substitution, the simplification process, and the final determination of whether -7 is a solution or not. By the end of this process, we will have not only found the correct answer but also gained a deeper understanding of how to solve equations and verify solutions.

Testing Equation A: 5x + 11 = -24

Let’s start with the first equation, 5x + 11 = -24. To determine if -7 is a solution, we'll substitute -7 for x in the equation. This gives us:

5(-7) + 11 = -24

Now, we need to simplify the left side of the equation. According to the order of operations, we perform multiplication before addition. So, we multiply 5 by -7:

-35 + 11 = -24

Next, we add -35 and 11. Remember that adding a negative number is like subtracting:

-24 = -24

We have arrived at a statement where the left side (-24) is equal to the right side (-24). This means that when we substituted -7 for x, the equation held true. Therefore, -7 is a solution to the equation 5x + 11 = -24.

This process illustrates the fundamental principle of verifying solutions: substituting the proposed solution into the equation and checking for equality. In this case, the equality holds, confirming that -7 is indeed a solution. However, we're not done yet! We need to test the other equations to see if any of them also have -7 as a solution. It's possible that more than one equation could satisfy the condition, or it’s possible that only this one does. To be thorough, we must analyze each equation individually.

In the subsequent sections, we will apply the same method to equations B, C, and D. We will substitute -7 for x, simplify, and check for equality. This will not only help us find the correct answer but also reinforce our understanding of equation solving and solution verification. So, let’s move on to the next equation and continue our investigation!

Testing Equation B: 4x - 23 = 5

Now, let's examine the second equation, 4x - 23 = 5. As before, we will substitute -7 for x and simplify to see if the equation remains true. Substituting -7 for x gives us:

4(-7) - 23 = 5

Following the order of operations, we first perform the multiplication. 4 multiplied by -7 is -28:

-28 - 23 = 5

Next, we subtract 23 from -28. This is the same as adding -23 to -28:

-51 = 5

In this case, the left side of the equation (-51) is not equal to the right side (5). This tells us that -7 is not a solution to the equation 4x - 23 = 5. When we substitute -7 for x, the equation does not hold true; the two sides are not balanced.

This result is important because it narrows down the possibilities. We now know that -7 is a solution to equation A, but not to equation B. This highlights the fact that a solution is specific to an equation; a value that solves one equation might not solve another.

The process of testing each equation is crucial in problems like these. It's not enough to find one solution; we need to verify that it's the correct one and that no other options also work. This systematic approach ensures accuracy and a thorough understanding of the problem.

In the following sections, we will continue this process by testing equations C and D. We will follow the same steps: substitute -7 for x, simplify, and check for equality. This will lead us to the final answer and further reinforce our equation-solving skills.

Testing Equation C: -x + 3 = -4

Moving on to equation C, -x + 3 = -4, we again substitute -7 for x. This is where it's crucial to pay attention to the negative signs. Substituting -7 for x gives us:

-(-7) + 3 = -4

Notice that we have a negative sign in front of the x, and we are substituting -7. The negative of a negative number is a positive number. So, -(-7) becomes +7:

7 + 3 = -4

Now, we add 7 and 3:

10 = -4

Here, the left side of the equation (10) is clearly not equal to the right side (-4). Therefore, -7 is not a solution to the equation -x + 3 = -4. This is another instance where the substitution does not result in a true statement.

This step further reinforces the importance of careful attention to signs and the rules of arithmetic. A seemingly simple negative sign can change the entire outcome of the equation. Accuracy in these details is paramount for successful equation solving.

We are now one step closer to the solution. We know that -7 is a solution to equation A, but not to equations B and C. This leaves us with only one equation left to test: equation D. By testing each equation systematically, we are building a solid foundation for solving algebraic problems. We're not just looking for the answer; we're understanding the process.

In the next section, we will test equation D and, hopefully, arrive at the final answer with confidence.

Testing Equation D: -3x - 8 = -29

Finally, let's test the last equation, -3x - 8 = -29. We follow the same procedure: substitute -7 for x and simplify. Substituting -7 gives us:

-3(-7) - 8 = -29

First, we perform the multiplication. -3 multiplied by -7 is 21 (a negative times a negative is a positive):

21 - 8 = -29

Next, we subtract 8 from 21:

13 = -29

In this case, the left side of the equation (13) is not equal to the right side (-29). Thus, -7 is not a solution to the equation -3x - 8 = -29. This is the third equation that does not have -7 as a solution.

With this final test, we have completed our analysis of all four equations. We systematically substituted -7 for x in each equation and checked for equality. We found that only equation A held true when -7 was substituted.

This comprehensive approach is essential for solving problems accurately. By testing each option, we eliminate the possibility of overlooking the correct answer or choosing an incorrect one. It also solidifies our understanding of the relationship between equations and their solutions.

In the final section, we will summarize our findings and state the solution to the problem.

Conclusion: The Solution

After carefully analyzing all four equations, we have determined that -7 is a solution to only one of them. Let's recap our findings:

• Equation A: 5x + 11 = -24. When we substituted -7 for x, the equation held true: -24 = -24.
• Equation B: 4x - 23 = 5. Substituting -7 resulted in -51 = 5, which is false.
• Equation C: -x + 3 = -4. Substituting -7 resulted in 10 = -4, which is false.
• Equation D: -3x - 8 = -29. Substituting -7 resulted in 13 = -29, which is false.

Therefore, the equation that has a solution of -7 is A. 5x + 11 = -24. This was the only equation where substituting -7 for x resulted in a true statement.

This problem highlights the importance of understanding what a solution to an equation means and how to verify it. The process of substitution and simplification is fundamental to algebra and is used extensively in more advanced mathematical concepts.

We hope this step-by-step explanation has clarified the process and provided you with a solid understanding of how to solve similar problems. Remember, the key is to be methodical, pay attention to details (especially signs), and follow the order of operations.

For further exploration of algebra and equation solving, you might find helpful resources at websites like Khan Academy's Algebra Section.