What Is The Value Of (x – y) (x – y) if xy = 12 and x2 + y2 = 25?
When given the equations xy = 12 and x^2 + y^2 = 25, we are asked to find the value of (x – y) (x – y). To solve this problem, let’s break it down step by step.
First, let’s find the values of x and y. By rearranging the equation xy = 12, we can express y in terms of x as y = 12/x. Substituting this into the equation x^2 + y^2 = 25, we get x^2 + (12/x)^2 = 25.
Simplifying further leads us to a quadratic equation: x^4 – 25x^2 +144 = 0. We can solve this equation using factoring or the quadratic formula. After finding the values of x, we can substitute them back into one of the original equations to find their corresponding values for y.
Once we have determined both values for x and y, we can calculate (x – y) (x – y) by substituting these values into the expression. This will give us our final answer.
Solving for x and y: Introduction
In this section, we’ll dive into solving for the variables x and y in the given equations. By understanding how to manipulate these equations, we can determine the value of (x – y) (x – y). Let’s get started!
Determining the value of (x – y) (x – y)
To find the value of (x – y) (x – y), we need to first solve for x and y using the given information. The two equations we have are xy = 12 and x^2 + y^2 = 25.
To begin, let’s rearrange the first equation to solve for one variable in terms of the other. We can isolate either x or y by dividing both sides by the variable not being isolated. For example, if we divide both sides by x, we get:
y = 12/x
Now, let’s substitute this expression for y into the second equation:
x^2 + (12/x)^2 = 25
Next, we simplify and rearrange this equation to solve for x. Multiplying through by x^2 gives us:
x^4 + 144 = 25x^2
Bringing all terms to one side yields a quartic equation:
x^4 – 25x^2 + 144 = 0
At this point, solving this quartic equation becomes a bit more involved. We can use various methods such as factoring or numerical approximation techniques like Newton’s method.
Once we find values for x, substituting them back into any of our original equations will give us corresponding values for y.
Understanding the given equations
The given equations provide constraints on how x and y relate to each other. The equation xy = 12 tells us that their product is equal to 12. This means that their values are inversely proportional; as one variable increases, the other decreases.
The second equation x^2 + y^2 = 25 represents a circle centered at the origin with a radius of 5. Any point (x, y) lying on this circle satisfies this equation. By combining these two equations, we can find specific points that satisfy both conditions.
Understanding the given equations helps us visualize the relationship between x and y and provides insight into how to solve for them.
Now that we have a better understanding of how to approach solving for x and y, let’s move on to actually finding their values in order to determine (x – y) (x – y). Applying the Substitution Method
The substitution method is a powerful tool in solving algebraic equations. In this case, we’ll use it to find the value of (x – y)(x – y) when given that xy = 12 and x^2 + y^2 = 25.
To begin, let’s solve for one variable in terms of the other using one of the equations. In this case, we can solve for y in terms of x using the equation xy = 12. Dividing both sides by x, we get:
y = 12/x
Now that we have an expression for y, we can substitute it into the second equation to eliminate one variable. Substituting (12/x) for y in the equation x^2 + y^2 = 25
Using the substitution method allowed us to solve for both variables and find all possible solutions for (x-y)(x-y) in a systematic manner.
In conclusion, by solving the given equations and substituting their solutions into the expression (x -y)(x-y), we can determine its value. Let’s explore further to unveil more about this intriguing mathematical problem.
