Equations With The Same Solution For X
#h1 Finding Equivalent Equations for x
In the realm of mathematics, particularly algebra, a fundamental task involves solving equations to determine the value(s) of unknown variables. Often, a single equation can be manipulated in various ways while maintaining the same solution set. This exploration delves into identifying equations that share an identical solution for the variable x as the given equation: $\frac{3}{5}(30 x-15)=72$. We will systematically analyze each provided option, employing algebraic techniques to simplify and compare them, ultimately pinpointing the three equations that yield the same x value.
To begin, let's solve the original equation for x. This will serve as our benchmark, allowing us to verify whether the solutions of the other equations coincide. The original equation is: $\frac3}{5}(30 x-15)=72$. The initial step involves distributing the fraction $\frac{3}{5}$ across the terms within the parentheses. This yields5} * 30x - \frac{3}{5} * 15 = 72$, which simplifies to $18x - 9 = 72$. Next, we isolate the term containing x by adding 9 to both sides of the equation{18} = \frac{81}{18}$, which simplifies to $x = 4.5$. Therefore, the solution to the original equation is $x = 4.5$. Now, we will evaluate each of the provided options to identify those that share this solution.
The core of solving equations lies in the ability to manipulate them while preserving the fundamental equality. This often involves applying operations to both sides of the equation, such as adding, subtracting, multiplying, or dividing by the same quantity. The distributive property plays a crucial role when dealing with expressions enclosed in parentheses, while the concept of inverse operations allows us to isolate the variable of interest. Equations that appear different on the surface may, in fact, be equivalent, sharing the same solution set. This principle is central to our task of identifying equations with the same x value as the original equation.
#h2 Analyzing the Provided Options
Now that we have determined the solution to the original equation ($x = 4.5$), we can proceed to examine each of the given options to ascertain which ones have the same solution. This involves solving each equation individually and comparing the resulting x value with our benchmark.
Option 1: $18 x-15=72$
To solve this equation for x, we first isolate the term containing x by adding 15 to both sides: $18x - 15 + 15 = 72 + 15$, which simplifies to $18x = 87$. Then, we divide both sides by 18: $\frac{18x}{18} = \frac{87}{18}$, resulting in $x = \frac{87}{18} = 4.8333...$. Since this value of x does not match the solution of the original equation (4.5), this option is not one of the correct choices.
Option 2: $50 x-25=72$
To solve this equation, we begin by adding 25 to both sides: $50x - 25 + 25 = 72 + 25$, which simplifies to $50x = 97$. Next, we divide both sides by 50: $\frac{50x}{50} = \frac{97}{50}$, resulting in $x = \frac{97}{50} = 1.94$. This value of x is also different from the solution of the original equation, so this option is not a correct choice.
Option 3: $18 x-9=72$
This equation is particularly interesting because it resembles a simplified form of the original equation. We add 9 to both sides: $18x - 9 + 9 = 72 + 9$, which simplifies to $18x = 81$. Dividing both sides by 18 gives us: $\frac{18x}{18} = \frac{81}{18}$, which simplifies to $x = \frac{81}{18} = 4.5$. This value of x matches the solution of the original equation, indicating that this option is one of the correct choices.
Option 4: $3(6 x-3)=72$
To solve this equation, we first distribute the 3 across the terms within the parentheses: $3 * 6x - 3 * 3 = 72$, which simplifies to $18x - 9 = 72$. This is the same equation as Option 3, and as we have already established, its solution is $x = 4.5$. Therefore, this option is also a correct choice.
Option 5: $x=4.5$
This option directly states the solution for x. Since it matches the solution of the original equation, this is also a correct choice.
By meticulously solving each equation, we have identified the values of x for each option. This process underscores the importance of algebraic manipulation in maintaining equivalence and reveals how seemingly different equations can lead to the same solution. The distributive property, the addition/subtraction property of equality, and the multiplication/division property of equality are fundamental tools in this endeavor. Recognizing and applying these principles effectively is crucial for solving equations accurately and efficiently.
#h3 Conclusion and Final Answer
In conclusion, we were tasked with identifying three equations that share the same solution for x as the original equation, $\frac{3}{5}(30 x-15)=72$. After solving the original equation and obtaining $x = 4.5$, we systematically analyzed each of the provided options.
Through this process, we determined that the following three equations have the same value of x as the original equation:
- Option 3: $18x - 9 = 72$ (Solution: $x = 4.5$)
- Option 4: $3(6x - 3) = 72$ (Solution: $x = 4.5$)
- Option 5: $x = 4.5$ (Directly states the solution)
Therefore, the correct answers are Options 3, 4, and 5. This exercise highlights the significance of understanding equivalent equations and the application of algebraic principles to solve for unknown variables. The ability to manipulate equations while preserving their solutions is a cornerstone of algebra and mathematical problem-solving in general.
