Problem 5 - Algebra and Trigonometry

Question

\(3-8=\) 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: Express y in terms of x

The second equation of the system is \(x + y = 0\). We can solve for \(y\) in terms of \(x\) by rewriting this equation as \(y = -x\). This expression will be used to substitute into the first equation.

Step 2: Substitute y into the first equation

Substitute \(y = -x\) into the first equation \(x^2 + y^2 = 8\). Thus, we have \(x^2 + (-x)^2 = 8\), which simplifies to \(x^2 + x^2 = 8\).

Step 3: Simplify and solve for x

Simplify the equation from the previous step: \(2x^2 = 8\). Divide both sides by 2, which gives \(x^2 = 4\). Now, solve for \(x\) by taking the square root of both sides: \(x = 2\) or \(x = -2\).

Step 4: Find corresponding y values

Using \(y = -x\), substitute \(x = 2\) to find \(y = -2\). Then, using \(x = -2\), we have \(y = 2\).

Step 5: List all the solutions

The solutions to the system are the pairs \((x, y) = (2, -2)\) and \((x, y) = (-2, 2)\).

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations that have one or more variables in common. In mathematics, solving a system involves finding the values of the variables that satisfy all the equations simultaneously.
For the given system of equations:

• Equation 1: \(x^2 + y^2 = 8\)
• Equation 2: \(x + y = 0\)

We are tasked with finding values for \(x\) and \(y\) that make both equations true simultaneously.There are several methods to solve systems of equations such as substitution, elimination, and graphing. Here, the substitution method is used, where one variable is expressed in terms of another, and then substituted into the other equation to find a solution.

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They are important in mathematics because they introduce the concept of non-linearity.
In this exercise, after substituting \(y = -x\) into the first equation, we obtain:

• \(x^2 + (-x)^2 = 8\)
• This simplifies to \(2x^2 = 8\)
• Dividing both sides by 2, we get \(x^2 = 4\)

Here, \(x^2 = 4\) is a quadratic equation. We can solve it by taking the square root of both sides, bearing in mind it produces two solutions:

• \(x = 2\) and \(x = -2\)

This reflects the property of quadratic equations, that they can often have two solutions.

Understanding Coordinate Pairs

Coordinate pairs, commonly written as \((x, y)\), are a way to represent points on a graph. These pairs denote the position of a point in a two-dimensional plane, where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate.
In our solved system of equations, once \(x\) was found as 2 or -2, we determined the corresponding \(y\) values using the relationship \(y = -x\). From the solutions:

• If \(x = 2\), then \(y = -2\).
• If \(x = -2\), then \(y = 2\).

These give us the coordinate pairs \((2, -2)\) and \((-2, 2)\). These pairs are not just numbers; they represent exact points that satisfy both equations in the system. By locating these points on a graph, one could visualize the intersections of the curves represented by these equations.