Problem 5 - Algebra and Trigonometry

Question

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {x+y=0}\end{array}\right.\)

Step-by-Step Solution

Answer

The solutions are \((2, -2)\) and \((-2, 2)\).

Step 1: Solve for One Variable

From the equation \(x + y = 0\), we can express \(y\) in terms of \(x\): \[ y = -x \]

Step 2: Substitute and Simplify

Substitute \(y = -x\) into the first equation \(x^2 + y^2 = 8\):\[x^2 + (-x)^2 = 8\]Simplify:\[x^2 + x^2 = 8\]which simplifies to:\[2x^2 = 8\]

Step 3: Solve for x

Divide both sides of the equation \(2x^2 = 8\) by 2:\[x^2 = 4\]Take the square root of both sides:\[x = 2 \quad \text{or} \quad x = -2\]

Step 4: Solve for y

Using \(y = -x\), substitute the values of \(x\) to find \(y\):- If \(x = 2\), then \(y = -2\).- If \(x = -2\), then \(y = 2\).

Step 5: Verify the Solutions

Verify both pairs:\((2, -2)\) and \((-2, 2)\) satisfy the original equations:For \((2, -2)\):- \(2 + (-2) = 0\)- \(2^2 + (-2)^2 = 8\)For \((-2, 2)\):- \((-2) + 2 = 0\)- \((-2)^2 + 2^2 = 8\)Both pairs satisfy both equations.

Key Concepts

Understanding Systems of Equations Through Substitution

When tackling a set of equations like the one provided, you are dealing with a system of equations. A system of equations involves two or more equations that share common variables. In this case, our variables are \(x\) and \(y\). The goal here is to find the values of these variables that satisfy every equation in the system simultaneously.

One common method to solve systems of equations is the substitution method. This method involves expressing one variable in terms of another using one of the equations and then substituting this expression into the other equations. This process reduces the number of variables and allows you to solve the system more easily.

For example, in the equations \(x + y = 0\) and \(x^2 + y^2 = 8\), you can express \(y\) as \(y = -x\) using the first equation. Then, substitute \(y = -x\) into the second equation, resulting in a single equation in terms of \(x\). This conversion helps focus on one variable at a time, simplifying the process of finding the solution.

Solving Quadratic Equations by Simplification

Once you've substituted expressions to focus on a single variable, you often end up with a quadratic equation. Quadratic equations typically take the form \(ax^2 + bx + c = 0\). In our exercise, the equation simplified to \(2x^2 = 8\), which is a straightforward case.

A crucial step in solving quadratic equations is to isolate the \(x^2\) term first. Here, we divided both sides by 2, yielding \(x^2 = 4\). Next, to solve for \(x\), take the square root of both sides. Remember that taking a square root introduces a positive and a negative solution, giving \(x = 2\) or \(x = -2\).

These solutions indicate the x-values that satisfy the relationships given in the problem. Such steps illustrate the basic yet essential pattern of handling quadratics that emerge from substitution in systems of equations.

Verification of Solutions in Systems

The final step in solving a system of equations is to verify that your solutions are correct. Verification is essential to ensure that the solutions satisfy all original equations in the system.

To verify, take each solution pair (in this case, \((2, -2)\) and \((-2, 2)\)) and substitute these values back into the original equations. Check each equation:

• For \((2, -2)\), substitute into \(x + y = 0\): \(2 + (-2) = 0\) and into \(x^2 + y^2 = 8\): \(2^2 + (-2)^2 = 8\).
• For \((-2, 2)\), substitute: \((-2) + 2 = 0\) and \((-2)^2 + 2^2 = 8\).

Both pairs confirm the solutions because they satisfy both of the initial equations. Verifying solutions garners confidence that the found solutions correctly solve the system, reflecting the logical underpinnings of algebraic methods like substitution. This step is vital to check both the accuracy and completeness of your results.