91Ó°ÊÓ

The elimination method is a useful technique for solving systems of equations. It involves combining the equations in a way that eliminates one of the variables, making it easier to solve for the remaining variable. In this exercise, we use elimination to solve for the variables in the system:\[ \begin{cases} x^{2} + y^{2} = 20 \ x^{2} - y^{2} = -12 \end{cases}\]Here is a step-by-step overview:

• First, we label the equations for easy reference:
• Equation (1): \(x^{2} + y^{2} = 20\)
• Equation (2): \(x^{2} - y^{2} = -12\)
• We add Equation (1) and Equation (2):
• \(x^{2} + y^{2} + x^{2} - y^{2} = 20 + (-12)\)
• This simplifies to \(2x^{2} = 8\).

By cancelling out the \(y^{2}\) term, we reduced the system to a single equation in terms of \(x\). This makes it much simpler to solve.

A quadratic equation is a polynomial equation of degree 2, which means it includes terms up to \(x^2\). In this exercise, we encountered quadratic terms like \(x^{2}\) and \(y^{2}\). After using the elimination method, we reduced our system to:\[2x^{2} = 8\]We divided both sides by 2 to solve for \(x^{2}\):\[x^{2} = 4\]This shows that \(x^{2}\) is 4, which we can solve further by taking the square root. Quadratic equations like these often have two solutions because squaring either a positive or negative number results in the same value. So, \(x^{2} = 4\) has solutions \(x = 2\) and \(x = -2\).

Substitution is another technique for solving systems of equations. It involves solving one equation for a variable and then substituting that expression into the other equation. In this exercise, after finding \(x = \pm 2\), we substituted these values back into the original Equation (1) to find \(y\). Let's break it down:

• For \(x = 2\), we substitute into Equation (1):
• \(2^{2} + y^{2} = 20\)
• Simplifying, \(4 + y^{2} = 20\)
• Solving for \(y\), \(y^{2} = 16\)
• So, \(y = \pm 4\)
• This gives solutions \((2, 4)\) and \((2, -4)\).
• We do the same for \(x = -2\) and find the same \( y\) values, \(( -2, 4)\) and \( (-2, -4)\).

Thus, using substitution in conjunction with elimination ensures we find all potential solutions.

Square roots are a fundamental concept in solving quadratic equations. They reverse the process of squaring a number. In this exercise, after simplifying our equations using elimination and substitution, we had:\[x^{2} = 4\]To solve for \(x\), we took the square root of both sides. Remember, the square root of a number can be both positive and negative because both squared result in the original positive number:

• \( \sqrt{4} = 2\)
• and also \( -\sqrt{4} = -2\)

Thus, \( x = 2\) and \( x = -2\). We used these roots to find \(y\) in the subsequent steps. Similarly, solving \( y^{2} = 16\) results in \( y = 4\) and \( y = -4\). Understanding square roots helps in solving many algebraic problems, not just quadratics but also higher math levels.