Solving Linear Equations And Finding Equivalent Solutions

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In the realm of mathematics, particularly in algebra, solving linear equations is a fundamental skill. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The goal in solving a linear equation is to find the value of the variable that makes the equation true. This article delves into the process of solving linear equations, focusing on identifying equivalent equations—those that share the same solution. We will explore the steps involved in simplifying and solving a given equation, and then apply these principles to determine which among a set of options has the same solution. Let's embark on this mathematical journey to master the art of solving linear equations and understanding the concept of equivalent solutions.

Understanding the Given Equation

Let's consider the given equation:

−16p+37=49−21p-16p + 37 = 49 - 21p

This is a linear equation in one variable, p. To find the solution, we need to isolate p on one side of the equation. The process involves several algebraic manipulations, ensuring that the equality is maintained throughout. Our initial equation presents a balance, much like a weighing scale. Whatever operation we perform on one side, we must also perform on the other to keep the scale balanced. This principle is the cornerstone of solving equations.

The first step in solving this equation is to gather the terms containing p on one side and the constant terms on the other. We can achieve this by adding 21p to both sides of the equation. This operation eliminates the -21p term on the right side and moves the p term to the left side. Simultaneously, we subtract 37 from both sides to move the constant term to the right side. These two operations are crucial in isolating the variable p and simplifying the equation.

By performing these steps, we transform the equation into a more manageable form, making it easier to isolate the variable. Each step we take is guided by the principle of maintaining equality, ensuring that the solution remains unchanged. The goal is to simplify the equation without altering its fundamental nature, bringing us closer to the value of p that satisfies the original equation.

Solving the Equation

To solve the equation $-16p + 37 = 49 - 21p$, our first step is to consolidate the terms involving p on one side of the equation. We achieve this by adding 21p to both sides:

−16p+21p+37=49−21p+21p-16p + 21p + 37 = 49 - 21p + 21p

This simplifies to:

5p+37=495p + 37 = 49

Now, we need to isolate the term with p. To do this, we subtract 37 from both sides:

5p+37−37=49−375p + 37 - 37 = 49 - 37

This simplifies to:

5p=125p = 12

Finally, to solve for p, we divide both sides by 5:

5p5=125\frac{5p}{5} = \frac{12}{5}

Thus, we find the solution:

p=125=2.4p = \frac{12}{5} = 2.4

This value, p = 2.4, is the solution to the given equation. It represents the specific value that, when substituted back into the original equation, makes the equation true. Now that we have the solution, we can use it as a benchmark to determine which of the provided options has the same solution. The next step is to evaluate each option, substituting p = 2.4 and checking if the equation holds true. This process will reveal which equations are equivalent to the original one, sharing the same solution.

Evaluating the Options

Now that we have found the solution to the original equation, p = 2.4, we will substitute this value into each of the given options to determine which one(s) have the same solution. This process will help us identify the equations that are equivalent to the original equation.

Option A: $-14 + 6p = -9 - 6p$

Substitute p = 2.4 into the equation:

−14+6(2.4)=−9−6(2.4)-14 + 6(2.4) = -9 - 6(2.4)

−14+14.4=−9−14.4-14 + 14.4 = -9 - 14.4

0.4=−23.40.4 = -23.4

This statement is false, so option A does not have the same solution.

Option B: $-55 + 12p = 5p + 16$

Substitute p = 2.4 into the equation:

−55+12(2.4)=5(2.4)+16-55 + 12(2.4) = 5(2.4) + 16

−55+28.8=12+16-55 + 28.8 = 12 + 16

−26.2=28-26.2 = 28

This statement is also false, indicating that option B does not share the same solution as the original equation.

Option C: $2 + 1.25p = -3.75$

Substitute p = 2.4 into the equation:

2+1.25(2.4)=−3.752 + 1.25(2.4) = -3.75

2+3=−3.752 + 3 = -3.75

5=−3.755 = -3.75

This statement is false as well, meaning option C does not have the same solution.

Since none of the options yielded a true statement when p = 2.4 was substituted, it seems there might be an issue with the options provided or a potential error in the calculations. Let's re-examine each option to ensure accuracy in our evaluation.

Re-evaluating the Options and Correcting Errors

Upon reviewing the calculations, it's crucial to ensure accuracy in our assessment of each option. Let's re-evaluate each one, paying close attention to the arithmetic and algebraic steps.

Option A: $-14 + 6p = -9 - 6p$

Substitute p = 2.4 into the equation:

−14+6(2.4)=−9−6(2.4)-14 + 6(2.4) = -9 - 6(2.4)

−14+14.4=−9−14.4-14 + 14.4 = -9 - 14.4

0.4=−23.40.4 = -23.4

This remains false. Let's solve this equation for p to find its actual solution:

Add 6p to both sides:

−14+6p+6p=−9−6p+6p-14 + 6p + 6p = -9 - 6p + 6p

−14+12p=−9-14 + 12p = -9

Add 14 to both sides:

−14+12p+14=−9+14-14 + 12p + 14 = -9 + 14

12p=512p = 5

Divide by 12:

p=512p = \frac{5}{12}

So, the solution for option A is not 2.4.

Option B: $-55 + 12p = 5p + 16$

Substitute p = 2.4 into the equation:

−55+12(2.4)=5(2.4)+16-55 + 12(2.4) = 5(2.4) + 16

−55+28.8=12+16-55 + 28.8 = 12 + 16

−26.2=28-26.2 = 28

This is still false. Let's solve this equation for p:

Subtract 5p from both sides:

−55+12p−5p=5p−5p+16-55 + 12p - 5p = 5p - 5p + 16

−55+7p=16-55 + 7p = 16

Add 55 to both sides:

−55+7p+55=16+55-55 + 7p + 55 = 16 + 55

7p=717p = 71

Divide by 7:

p=717≈10.14p = \frac{71}{7} \approx 10.14

Thus, the solution for option B is also not 2.4.

Option C: $2 + 1.25p = -3.75$

Substitute p = 2.4 into the equation:

2+1.25(2.4)=−3.752 + 1.25(2.4) = -3.75

2+3=−3.752 + 3 = -3.75

5=−3.755 = -3.75

This remains false. Solving for p:

Subtract 2 from both sides:

2+1.25p−2=−3.75−22 + 1.25p - 2 = -3.75 - 2

1.25p=−5.751.25p = -5.75

Divide by 1.25:

p=−5.751.25=−4.6p = \frac{-5.75}{1.25} = -4.6

The solution for option C is not 2.4 either.

It appears that none of the provided options have the same solution as the original equation. This could indicate an error in the options themselves or in the initial problem statement. However, we have demonstrated the method for finding equivalent equations by solving for p in each case and comparing the solutions.

Conclusion

In conclusion, solving linear equations is a critical skill in mathematics. This article has provided a detailed walkthrough of how to solve a linear equation and how to identify equivalent equations. We started by solving the given equation, $-16p + 37 = 49 - 21p$, and found the solution to be p = 2.4. We then evaluated each of the provided options by substituting this value and checking for equality. Additionally, we demonstrated how to solve each option independently to find their respective solutions.

It is important to note that, upon careful re-evaluation, none of the options had the same solution as the original equation. This highlights the importance of accuracy in mathematical problem-solving and the necessity of verifying solutions. While we did not find an exact match in this case, the process we followed is the key to identifying equivalent equations. By understanding how to manipulate equations algebraically and solve for the variable, one can effectively determine if two equations are equivalent.

This exercise underscores the significance of mastering the fundamentals of algebra. The ability to solve linear equations and recognize equivalent forms is not only essential for academic success but also for various real-world applications. By consistently practicing and applying these principles, one can build confidence and proficiency in mathematics.