Solving Equations An In-Depth Guide To Finding Equivalent Equations
In the realm of algebra, solving equations is a fundamental skill. This article delves into the process of finding equations with the same solution for x as the given equation: $\frac{3}{5}(30x - 15) = 72$. We will explore the step-by-step method to solve for x, and then identify the equivalent equations from the provided options. This exploration will not only solidify your understanding of solving linear equations but also enhance your ability to recognize equivalent forms, a crucial skill in mathematical problem-solving.
Step 1 Simplifying the Given Equation
Our main goal here is to isolate x on one side of the equation. Let's start by simplifying the given equation: $\frac{3}{5}(30x - 15) = 72$. The first step involves distributing the $\frac{3}{5}$ across the terms inside the parentheses. This means we multiply both $30x$ and $-15$ by $\frac{3}{5}$. This distribution is a key algebraic manipulation that allows us to remove the parentheses and simplify the equation into a more manageable form. By carefully applying the distributive property, we set the stage for further simplification and ultimately, the solution for x.
When we multiply $\frac3}{5}$ by $30x$, we get5} * 30x = 18x$. Next, we multiply $\frac{3}{5}$ by $-15$, which yields{5} * -15 = -9$. So, after distributing, our equation looks like this: $18x - 9 = 72$. This simplified form is much easier to work with. We have successfully eliminated the fraction and parentheses, bringing us closer to isolating x. This initial simplification is a critical step in solving any algebraic equation, as it transforms the equation into a form where we can apply basic operations to isolate the variable.
Step 2 Isolating the Term with x
Now that we have simplified the equation to $18x - 9 = 72$, the next step is to isolate the term containing x, which is $18x$. To do this, we need to get rid of the $-9$ on the left side of the equation. We can achieve this by performing the inverse operation, which is adding $9$ to both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain the equality. This principle is a cornerstone of algebraic manipulation and ensures that the equation remains balanced throughout the solving process.
Adding $9$ to both sides, we get: $18x - 9 + 9 = 72 + 9$. This simplifies to $18x = 81$. We have now successfully isolated the term with x on one side of the equation. This step is crucial because it brings us closer to our ultimate goal of finding the value of x. By isolating the term, we are setting up the final step of dividing both sides by the coefficient of x. This process demonstrates the power of inverse operations in algebra, allowing us to systematically unravel the equation and solve for the unknown variable.
Step 3 Solving for x
We've reached the stage where we have $18x = 81$. To finally solve for x, we need to isolate x completely. Since x is being multiplied by $18$, we perform the inverse operation, which is dividing both sides of the equation by $18$. This will give us the value of x. Dividing both sides by the coefficient of x is a standard technique in algebra and is the final step in isolating the variable.
Dividing both sides by $18$, we get: $\frac{18x}{18} = \frac{81}{18}$. This simplifies to $x = \frac{81}{18}$. We can further simplify the fraction $\frac{81}{18}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $9$. This simplification gives us $x = \frac{9}{2}$. Converting this improper fraction to a decimal, we find that $x = 4.5$. Therefore, the solution to the equation $\frac{3}{5}(30x - 15) = 72$ is $x = 4.5$. This final step demonstrates the importance of simplification in algebra, as it allows us to express the solution in its simplest and most easily understood form.
Step 4 Identifying Equivalent Equations
Now that we know the value of x is $4.5$, we need to determine which of the given equations also have the solution $x = 4.5$. This involves substituting $x = 4.5$ into each equation and checking if the equation holds true. This process highlights the concept of equivalent equations – equations that, despite their different appearances, have the same solution set. Identifying equivalent equations is a valuable skill in mathematics, as it allows us to choose the form that is most convenient for a particular problem or context.
Let's examine each option:
-
Substitute $x = 4.5$: $18(4.5) - 15 = 81 - 15 = 66$. Since $66$ is not equal to $72$, this equation does not have the same solution.
-
Substitute $x = 4.5$: $50(4.5) - 25 = 225 - 25 = 200$. Since $200$ is not equal to $72$, this equation does not have the same solution.
-
Substitute $x = 4.5$: $18(4.5) - 9 = 81 - 9 = 72$. This equation holds true for $x = 4.5$, so it is an equivalent equation.
-
First, distribute the $3$: $24x - 9 = 72$. Now, substitute $x = 4.5$: $24(4.5) - 9 = 108 - 9 = 99$. Since $99$ is not equal to $72$, this equation does not have the same solution.
-
This is simply the solution we found, so it is trivially equivalent.
Step 5 Selecting the Equivalent Equations
Based on our analysis, the equations that have the same value of x as $\frac{3}{5}(30x - 15) = 72$ are:
To find the third equivalent equation, let’s manipulate the original equation $\frac{3}{5}(30x - 15) = 72$:
Multiply both sides of the equation $18x - 9 = 72$ by $5/3$:
So, the third equation is $30x - 15 = 120$
Therefore, the three equivalent equations are:
This detailed step-by-step solution not only answers the question but also provides a thorough understanding of how to solve linear equations and identify equivalent forms. The ability to solve equations and recognize equivalencies is a crucial skill in mathematics and is applicable in various fields, from engineering to economics. By mastering these concepts, you empower yourself to tackle a wide range of problems and challenges.
In this comprehensive guide, we have meticulously walked through the process of solving the equation $\frac{3}{5}(30x - 15) = 72$ and identifying equivalent equations. We began by simplifying the original equation, isolating the term with x, and ultimately solving for x. This process involved the application of fundamental algebraic principles such as the distributive property, inverse operations, and simplification of fractions. Each step was explained in detail to provide a clear understanding of the underlying logic and techniques. Furthermore, we demonstrated how to verify the equivalence of equations by substituting the solution back into the equations and checking for equality. This is a crucial step in ensuring the correctness of our solutions and reinforcing the concept of equivalent equations.
We identified the three equivalent equations: $18x - 9 = 72$, $x = 4.5$, and $30x - 15 = 120$. This exercise not only provided a solution to the specific problem but also highlighted the importance of understanding different forms of equations and their relationships. The ability to manipulate equations and recognize equivalent forms is a powerful tool in mathematical problem-solving. It allows us to approach problems from different angles, choose the most efficient method, and gain deeper insights into the underlying mathematical structures.
The skills and concepts covered in this article are fundamental to algebra and have wide-ranging applications in mathematics and other disciplines. By mastering these skills, you will be well-equipped to tackle more complex problems and excel in your mathematical pursuits. Remember, practice is key to mastering any mathematical skill, so continue to solve equations, explore different techniques, and challenge yourself with increasingly complex problems. With dedication and perseverance, you can unlock the power of algebra and apply it to solve real-world problems and advance your knowledge in various fields.
