Solving the equation 58: 2x^2 – 9x^2; 5 – 3x + y + 6 is a fundamental task in mathematics, often presenting challenges, especially when dealing with intricate expressions. In this article, we will meticulously guide through the process of solving this equation step by step, exploring various methods and techniques to simplify it and ascertain the values of ‘x’ and ‘y’ that satisfy it. By the conclusion of this article, you will possess a robust comprehension of equation-solving, empowering you to confront similar problems with confidence.
Understanding the Equation 58: 2x^2 – 9x^2; 5 – 3x + y + 6
Before diving into the solution, let’s closely examine the given equation: 58: 2x^2 – 9x^2; 5 – 3x + y + 6. This equation comprises three distinct terms: 2x^2, -9x^2, and 5 – 3x + y + 6. Our primary goal is to determine the values of ‘x’ and ‘y’ that validate this equation and render it true.
Combining Like Terms
To simplify the equation, we commence by consolidating like terms. In this instance, we have two terms containing x^2, namely 2x^2 and -9x^2. By merging these terms, we obtain -7x^2. Consequently, the equation transforms into 58: -7x^2; 5 – 3x + y + 6.
Isolating Variables
Isolating ‘x’
To proceed towards finding the values of ‘x,’ our aim is to isolate ‘x’ on one side of the equation. Let’s relocate all terms containing ‘x’ to the left side and constants to the right side:
-7x^2 = 58 – (5 + 6) + y
Isolating ‘y’
Similarly, to ascertain the value of ‘y,’ we need to isolate it on one side of the equation:
y = 58 – (5 + 6) – 7x^2
Using the Quadratic Formula
Given the quadratic term (-7x^2) in the equation, we can explore another method to find the solutions for ‘x’ utilizing the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for ‘x’ are given by:
x = (-b ± √(b^2 – 4ac)) / 2a
In our equation, a = -7, b = 0, and c = 58 – (5 + 6) + y. Substituting these values into the quadratic formula, we obtain:
x = (± √(0 – 4*(-7)(58 – (5 + 6) + y))) / 2(-7)
Checking the Solutions
Upon acquiring the values for ‘x’ using either method, it is crucial to validate these solutions by substituting them back into the original equation. This step ensures the equality of both sides of the equation, thereby confirming the accuracy of our solutions.
Graphical Representation
An alternative approach to visualize the solutions is through graphical representation. By plotting the equation on a graph and identifying the points where it intersects the x-axis, we can determine the solutions for ‘x.’ Additionally, graphical representation offers valuable insights into the behavior of the equation.
Conclusion
Equation-solving is an indispensable skill with applications across various mathematical domains and beyond. In this article, we have effectively addressed the equation 58: 2x^2 – 9x^2; 5 – 3x + y + 6 employing diverse methods, such as isolating variables and utilizing the quadratic formula. By comprehending these techniques, you can confidently tackle similar equations and arrive at precise solutions.