Solving For X When Y Equals 0 In The Equation Y = 3[x - (3 - X)] - 5(x + 1)
In mathematics, finding the values of variables that satisfy a given equation is a fundamental problem. This article will delve into the process of determining the values of x for which the equation y = 3[x - (3 - x)] - 5(x + 1) equals zero. This involves simplifying the equation, isolating the variable x, and ultimately solving for its value(s). This problem falls under the category of algebra, specifically dealing with linear equations. Understanding how to solve such equations is crucial for various applications in mathematics, science, and engineering.
We will begin by carefully simplifying the given equation, step by step, ensuring that each operation is mathematically sound. This process will involve distributing constants, combining like terms, and rearranging the equation to isolate x. By following these steps diligently, we can transform the original equation into a simpler form that allows us to readily identify the value(s) of x that satisfy the condition y = 0. The final solution will represent the x-value(s) that make the equation true, providing a clear and concise answer to the problem.
The problem at hand is to find all values of x such that y = 0, where the equation is given by:
y = 3[x - (3 - x)] - 5(x + 1)
This equation represents a linear relationship between x and y. Our goal is to determine the specific values of x that will make y equal to zero. This is a common type of problem in algebra and is essential for understanding how to solve equations and find solutions.
To solve this problem, we will need to simplify the equation by expanding the terms, combining like terms, and then isolating x. This process will involve a series of algebraic manipulations that will lead us to the solution. The final answer will be the value(s) of x that satisfy the equation when y is equal to zero. Let's embark on the journey to simplify and solve this equation, step by step, ensuring a clear and understandable solution process.
To find the values of x for which y = 0, we need to solve the equation:
0 = 3[x - (3 - x)] - 5(x + 1)
Step 1: Simplify the expression inside the brackets
First, we'll simplify the expression inside the square brackets:
x - (3 - x) = x - 3 + x = 2x - 3
This step involves distributing the negative sign and combining the x terms. By carefully applying the distributive property and combining like terms, we have successfully simplified the expression within the brackets. This simplification is crucial as it allows us to proceed with the next steps of the solution process, where we will further expand and simplify the equation.
Step 2: Substitute the simplified expression back into the equation
Now, substitute this back into the original equation:
0 = 3(2x - 3) - 5(x + 1)
By replacing the original expression inside the brackets with its simplified form, we have created a more manageable equation. This substitution is a key step in the process of solving for x, as it reduces the complexity of the equation and allows us to proceed with the next steps of expansion and simplification. The goal is to isolate x on one side of the equation, and this substitution brings us closer to that goal.
Step 3: Distribute the constants
Next, distribute the constants 3 and -5:
0 = 6x - 9 - 5x - 5
In this step, we have applied the distributive property to expand the equation further. This involves multiplying the constants outside the parentheses by each term inside the parentheses. The result is an equation with individual terms that can be easily combined and simplified. This step is crucial in the process of isolating x and finding its value(s). By carefully distributing the constants, we have prepared the equation for the next step, which is combining like terms.
Step 4: Combine like terms
Combine the x terms and the constant terms:
0 = (6x - 5x) + (-9 - 5)
0 = x - 14
This step involves identifying and combining terms that have the same variable (x) and terms that are constants. By grouping these terms together, we simplify the equation and make it easier to solve for x. The process of combining like terms is a fundamental skill in algebra, and it allows us to reduce the complexity of equations and isolate the variable of interest. In this case, combining like terms has brought us closer to the solution by simplifying the equation to a more manageable form.
Step 5: Isolate x
To isolate x, add 14 to both sides of the equation:
x = 14
By adding 14 to both sides of the equation, we have successfully isolated x and found its value. This is the final step in the solution process, where we have determined the specific value of x that satisfies the equation when y is equal to zero. The value x = 14 is the solution to the problem, and it represents the point where the line defined by the equation intersects the x-axis. This solution can be verified by substituting x = 14 back into the original equation and confirming that y equals zero.
The value of x for which y = 0 is:
x = 14
Therefore, the solution to the equation 3[x - (3 - x)] - 5(x + 1) = 0 is x = 14. This means that when x is equal to 14, the value of y will be zero. This solution can be verified by substituting x = 14 back into the original equation and confirming that the result is indeed zero. This final answer provides a clear and concise solution to the problem, demonstrating the application of algebraic principles to solve for the value of a variable.
In this article, we have successfully found the value of x for which the equation y = 3[x - (3 - x)] - 5(x + 1) equals zero. By following a step-by-step approach, we simplified the equation, combined like terms, and isolated x to find the solution x = 14. This process demonstrates the fundamental principles of algebra and how they can be applied to solve linear equations. Understanding these principles is crucial for various mathematical and scientific applications.
The ability to solve equations is a fundamental skill in mathematics and is essential for problem-solving in various fields. By mastering these techniques, we can tackle more complex problems and gain a deeper understanding of mathematical concepts. The solution x = 14 represents a specific point on the graph of the equation, where the line intersects the x-axis. This visual representation can further enhance our understanding of the solution and its significance. In conclusion, this article has provided a comprehensive guide to solving the given equation and finding the value of x that satisfies the condition y = 0.
