Verifying Solutions For Equations Is X=-3 A Solution For 2x + 5 = -1

by Scholario Team 69 views

Hey guys! Today, we're diving into the world of algebra to check if x = -3 is the solution for the equation 2x + 5 = -1. We'll walk through the steps to solve this equation and confirm whether x = -3 truly fits the bill. So, grab your pencils, and let's get started!

Step-by-Step Verification

Let's begin by understanding the problem. We have the equation 2x + 5 = -1, and we want to verify if x = -3 is the solution. To do this, we need to follow a systematic approach that will lead us to a clear conclusion.

1. Substitute x with -3

The first step in verifying the solution is to substitute x with -3 in the given equation. This means we'll replace every instance of x in the equation 2x + 5 = -1 with -3. Our equation will then look like this:

2(-3) + 5 = -1

This substitution allows us to evaluate the left-hand side of the equation and see if it equals the right-hand side, which is -1.

2. Simplify the Equation

Next up, we need to simplify the equation by performing the arithmetic operations. Start by multiplying 2 by -3:

2 * (-3) = -6

Now our equation looks like this:

-6 + 5 = -1

Now, let's add -6 and 5:

-6 + 5 = -1

So, the simplified equation is:

-1 = -1

This result is crucial because it shows whether the left-hand side of the equation equals the right-hand side when x = -3.

3. Confirm the Solution

Now, let's analyze our result. We've simplified the equation to -1 = -1. This statement is true, meaning that when we substitute x = -3 into the original equation, the equation holds. This confirms that x = -3 is indeed a solution to the equation 2x + 5 = -1.

Solving the Equation

Alright, so we've seen how to verify a solution. But what if we wanted to find the solution ourselves? Let's walk through the steps to solve the equation 2x + 5 = -1 from scratch.

  1. Isolate the Term with x: Our first goal is to isolate the term with x on one side of the equation. To do this, we need to get rid of the +5 on the left side. We can do this by subtracting 5 from both sides of the equation:

    2x + 5 - 5 = -1 - 5

    This simplifies to:

    2x = -6

  2. Solve for x: Now that we have 2x = -6, we need to isolate x. To do this, we'll divide both sides of the equation by 2:

    2x / 2 = -6 / 2

    This simplifies to:

    x = -3

  3. Verify the Solution: As we demonstrated earlier, we can verify the solution by substituting x = -3 back into the original equation:

    2(-3) + 5 = -1

    -6 + 5 = -1

    -1 = -1

    Since the equation holds true, our solution x = -3 is correct.

Correct Alternative

Given the question and the steps we've covered, the correct alternative is:

A) Substitute x by -3 and simplify the equation.

This is the core of verifying whether a given value is a solution to an equation. By substituting the value and simplifying, we can see if the equation holds true.

In Depth Explanation

To truly grasp the process of verifying solutions, let's dive deeper into why these steps work. When we substitute a value for x, we're essentially testing whether that value makes the equation a true statement. An equation is like a balanced scale; both sides must be equal for the scale to be balanced.

Understanding Substitution

Substitution is the act of replacing a variable (like x) with a specific value. In our case, we replaced x with -3. This allows us to transform the equation from an algebraic expression into a numerical one that we can evaluate.

Simplifying Expressions

Simplifying the equation involves performing the arithmetic operations in the correct order (PEMDAS/BODMAS - Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This process reduces the equation to its simplest form, making it easier to see if both sides are equal.

The Logic of Equality

The fundamental principle here is that for a value to be a solution, it must satisfy the equation. This means that when we substitute the value and simplify, the left-hand side of the equation must equal the right-hand side. If they don't, the value is not a solution.

Why This Matters

Understanding how to verify solutions is a crucial skill in algebra. It helps you:

  • Check Your Work: You can use this method to ensure your solutions are correct.
  • Avoid Mistakes: By verifying, you can catch errors you might have made while solving the equation.
  • Build Confidence: Knowing you can verify your answers builds confidence in your problem-solving abilities.

Real-World Applications

Algebraic equations aren't just abstract math problems; they have real-world applications. For example:

  • Finance: Calculating interest rates or loan payments.
  • Physics: Solving for velocity, acceleration, or distance.
  • Engineering: Designing structures and systems.

The ability to solve and verify equations is a valuable skill in many fields.

Additional Examples

To solidify our understanding, let's look at a couple more examples.

Example 1: Verifying x = 2 for 3x - 4 = 2

  1. Substitute: Replace x with 2: 3(2) - 4 = 2
  2. Simplify: 6 - 4 = 2, which simplifies to 2 = 2
  3. Confirm: Since 2 = 2, x = 2 is a solution.

Example 2: Verifying x = -1 for 4x + 7 = 3

  1. Substitute: Replace x with -1: 4(-1) + 7 = 3
  2. Simplify: -4 + 7 = 3, which simplifies to 3 = 3
  3. Confirm: Since 3 = 3, x = -1 is a solution.

Common Mistakes to Avoid

When verifying solutions, here are some common mistakes to watch out for:

  • Incorrect Substitution: Make sure you replace every instance of x with the value.
  • Arithmetic Errors: Double-check your calculations, especially when dealing with negative numbers.
  • Order of Operations: Follow PEMDAS/BODMAS to ensure you simplify the equation correctly.

Conclusion

So, there you have it! We've walked through the steps to verify if x = -3 is the solution for the equation 2x + 5 = -1. We substituted, simplified, and confirmed that x = -3 indeed makes the equation true. Remember, this process is a fundamental skill in algebra and can help you check your work and build confidence in your problem-solving abilities. Keep practicing, and you'll become a pro at verifying solutions in no time!

Now you guys know how to verify solutions to equations like a champ! Keep up the great work, and remember, math can be fun when you break it down step by step. Happy problem-solving!