Finding Equations With Identical Solutions: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into the fascinating world of equations and discover which ones share the same solution. This is a crucial concept in algebra, so understanding it will seriously level up your math game. We'll break down the given equations step by step, solve them, and then identify the ones that have the same answer. Plus, we'll solve a bonus equation at the end! So, buckle up and let's get started!

Decoding Equations: The Basics

Before we jump into the problems, let's refresh our memory on what an equation is. An equation is a mathematical statement that asserts the equality of two expressions. It's like a balanced scale, where both sides must be equal. The goal in solving an equation is to find the value of the variable (usually represented by a letter like w) that makes the equation true. To do this, we use various algebraic manipulations, such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number. Remember, whatever you do to one side of the equation, you must do to the other to keep it balanced.

Now, let's look at the equations we are given. Each one presents a different form, involving addition, subtraction, multiplication, and division. Understanding how to handle each operation is key. For example, when isolating w, we use inverse operations. If a number is added to w, we subtract it from both sides; if w is multiplied by a number, we divide both sides by that number. In essence, the game is to isolate w on one side of the equation, revealing its value. So, as we examine each equation, we apply these principles. Consider the order of operations as well, which is crucial for simplifying complex expressions. This involves tackling parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). Getting this right keeps your calculations accurate. For the first equation $w - 4.95 = 2.30$, you will begin by isolating w. This involves adding 4.95 to both sides. For the second one, you will start by simplifying the right-hand side before isolating w. For the third one, you'll need to combine the numbers on the right side. And finally, for the fourth equation, you need to multiply both sides to eliminate the fraction. The goal is to perform each step carefully to uncover the value of w that satisfies each equation.

Analyzing Equation A: $w - 4.95 = 2.30$

Let's start by solving equation A: $w - 4.95 = 2.30$. Our goal is to isolate w. To do this, we need to get rid of the -4.95. Remember our golden rule: whatever we do to one side of the equation, we must do to the other to keep it balanced. So, we'll add 4.95 to both sides of the equation. This gives us:

$w - 4.95 + 4.95 = 2.30 + 4.95$

Simplifying this, we get:

$w = 7.25$

So, the solution to equation A is w = 7.25. Now, hold onto this value; we'll need it later to compare with the other equations!

Examining Equation B: $15w = 103.75 + 5$

Next up, let's tackle equation B: $15w = 103.75 + 5$. Here, w is being multiplied by 15. First, we'll simplify the right side of the equation. 103.75 + 5 equals 108.75. So, our equation becomes:

$15w = 108.75$

To isolate w, we need to divide both sides of the equation by 15. This gives us:

rac{15w}{15} = rac{108.75}{15}

Simplifying this, we get:

$w = 7.25$

Hey, look at that! The solution to equation B is also w = 7.25. This means equation B has the same solution as equation A. We are making progress!

Exploring Equation C: $25.4 + w = 23.25 + 9.4$

Now, let's move on to equation C: $25.4 + w = 23.25 + 9.4$. First, let's simplify the right side of the equation by adding 23.25 and 9.4. That gives us 32.65. So, our equation becomes:

$25.4 + w = 32.65$

To isolate w, we need to subtract 25.4 from both sides of the equation. This gives us:

$25.4 + w - 25.4 = 32.65 - 25.4$

Simplifying, we get:

$w = 7.25$

Awesome! Equation C also has the solution w = 7.25. So far, all three equations (A, B, and C) share the same solution. Remember that checking your work is important. Doing so can save a lot of time and potential confusion. For instance, to double-check the solution for C, plug 7.25 back into the original equation: 25.4 + 7.25 = 32.65. This confirms that our solution is correct. Moreover, in solving this, you are not only finding the value of w but also practicing core mathematical principles like the properties of equality and inverse operations. This builds a strong foundation for more complex mathematical problems later. Always remember that equations are like puzzles, and finding the right solution requires careful thought and a methodical approach.

Investigating Equation D: rac{w}{-0.25} = 14.5 + 14.5

Finally, let's analyze equation D: rac{w}{-0.25} = 14.5 + 14.5. First, we'll simplify the right side of the equation. 14.5 + 14.5 equals 29. So, our equation becomes:

rac{w}{-0.25} = 29

To isolate w, we need to multiply both sides of the equation by -0.25. This gives us:

rac{w}{-0.25} * -0.25 = 29 * -0.25

Simplifying, we get:

$w = -7.25$

Uh oh! The solution to equation D is w = -7.25. This is different from the solutions we found for equations A, B, and C. Therefore, equation D does not have the same solution as the other three.

Summary of Solutions

• Equation A: w = 7.25
• Equation B: w = 7.25
• Equation C: w = 7.25
• Equation D: w = -7.25

The Answer

So, which equations have the same solution? Based on our calculations, the equations that share the same solution are: A, B, and C.

Bonus Equation: Solving $0.14w = 0.70$

Now, let's solve the bonus equation: $0.14w = 0.70$. To isolate w, we need to divide both sides of the equation by 0.14. This gives us:

rac{0.14w}{0.14} = rac{0.70}{0.14}

Simplifying, we get:

$w = 5$

So, the solution to the bonus equation is w = 5. Not too shabby, right?

Wrapping Up

That's a wrap, guys! We've successfully navigated through multiple equations, identified those with the same solutions, and even solved a bonus equation. Remember, practice makes perfect. The more you work with equations, the more comfortable and confident you'll become. Keep practicing, keep learning, and keep that mathematical spirit alive! You've got this!

I hope this guide was helpful. Let me know if you have any questions. Happy solving!