91Ó°ÊÓ

Simultaneous equations are a set of equations containing multiple variables. The solution is the set of values that satisfy all the equations at once. In this case, we have two equations:

• \( x^2 + y^2 = 74 \)
• \( x^2 - y^2 = 24 \)

Both equations involve the same variables, \(x\) and \(y\).

Our goal is to find values of \(x\) and \(y\) such that both equations are true at the same time. A common technique to solve simultaneous equations is to add or subtract them in such a way that one of the variables gets eliminated.

After adding the given equations, it becomes possible to solve for \(x\) directly since we are left with an equation with only one variable.

The sum of squares is a concept where individual numbers (in this case, integers) are squared and then added together. For our problem:

• We have two numbers \(x\) and \(y\), and the sum of their squares is \(74\).

Written mathematically, this is represented as \(x^2 + y^2 = 74\).

This equation is crucial because it establishes a relationship between the two unknowns. By understanding that their squares add up to \(74\), we narrow down the possible values \(x\) and \(y\) can take.

It also shows the limitations: neither number can be too large, as their squares wouldn't total only \(74\). The sum of squares is used in various mathematical and practical fields, helping to understand and quantify the extent or distance of numerical values.

The final goal is to find integer solutions, meaning our answers for \(x\) and \(y\) should be whole numbers.

These integers, when squared and summed, must match the conditions given in our equations:

• The sum of squares \(x^2 + y^2 = 74\), and
• The difference of squares \(x^2 - y^2 = 24\).

Finding integer solutions involves not only solving the simultaneous equations but also verifying the solutions satisfy all conditions. We solved for \(x\) and found it to be \(7\), using it to find \(y\) as \(5\).

By carefully selecting operations on equations (addition, subtraction), we ensure our process stays algebraically sound while leading to whole number solutions. This is essential in mathematics when precision and accuracy are crucial, ensuring we meet all specified requirements.